problem solving quadratic expression

Quadratic Expressions

Quadratic expression is an expression with the variable with the highest power of 2. The word quadratic is derived from the word quad which means square. The expression should have the power of two and not higher or lower. Graphically a quadratic expression describes the path followed by a parabola, and it helps in finding the height and time of flight of a rocket.

In this mini-lesson, we will explore the world of quadratic expressions. You will get to learn how to solve quadratic expressions using the discriminant, simplifying quadratic expressions, factorizing expressions, quadratic expressions graphs, and other interesting facts around the topic.

Definition of Quadratic Expressions

An expression of the form ax 2 + bx + c, where a ≠ 0 is called a quadratic expression. In other words, any expression with the variables of the highest exponent or the expression's degree as 2 is a quadratic expression The standard form of a quadratic expression looks like this

Here are some examples of expressions:

Look at example 4, here a = 0. Therefore, it is not a quadratic expression. In fact, this is a linear expression. The remaining expressions are quadratic expression examples. The standard form of the standard expression in variable x is ax 2 + bx + c. But always remember that 'a' is a non-zero value and sometimes a quadratic expression is not written in its standard form.

Properties of Quadratic Expression

Listed below are a few important properties to keep in mind while identifying quadratic expressions.

• The expression is usually written in terms of x, y, z, or w. In the alphabets, the letters, in the end, are written for variables whereas the letters, in the beginning, are used for numbers.
• The variable 'a' in a quadratic expression raised to the power of 2 cannot be zero. If a = 0 then x 2 will be multiplied by zero and therefore, it would not be a quadratic expression anymore. Variable b or c in the standard form can be 0 but 'a' cannot.
• In a quadratic expression, it is not necessary that all the terms are positive. The standard form is written in a positive form. However, if the numbers are negative the term will also be negative.
• The terms in a quadratic expression are usually written with the power of 2 first, the power of 1 next, and the number in the end.

Ways of Writing a Quadratic Expression

Writing any expression or equation into a quadratic standard form, we need to follow these three methods.

• Move an expression by -1 if the equation starts with a negative value.
• Expand the terms and simply.
• Multiply the factors.

Mentioned below are a few examples:

Graphing a Quadratic Expressions

The graph of the quadratic expression ax 2 + bx + c = 0 can be obtained by representing the quadratic expression as a function y = ax 2 + bx + c. Further on solving and substituting values for x, we can obtain values of y, we can obtain numerous points. These points can be presented in the coordinate axis to obtain a parabola-shaped graph for the quadratic expression.

The point where the graph cuts the horizontal x-axis is the solution of the quadratic expression. These points can also be algebraically obtained by equalizing the y value to 0 in the function y = ax 2 + bx + c and solving for x. This is how the quadratic expression is represented on a graph. This curve is called a parabola .

Factorizing Quadratic Expressions

The factorizing of a quadratic expression is helpful for splitting it into two simpler expressions, which on multiplying, gives back the original quadratic expression. The aim of factorization is the break down the expression of higher degrees into expressions of lower degrees. Here the quadratic expression is of the second degree and is split into two simple linear expressions. The process of factorization of a quadratic equation includes the first step of splitting the x term, such that the product of the coefficients of the x-term is equal to the constant term. Further, the new expression after splitting the x term has four terms in the quadratic expression. Here some of the terms are taken commonly to obtain the final factors of the quadratic expression. Let us understand the process of factorizing a quadratic expression through an example.

Example 1: 4x - 12x 2

Given any quadratic expression, first, check for common factors, i.e. 4x and 12x 2

We can observe that 4x is a common factor. Let’s take that common factor from the quadratic expression.

4x - 12x 2 = 4x(1 - 3x)

Thus, by simplifying the quadratic expression, we get the expression, 4x - 12x 2 .

Quadratic Expressions Formula

Solving a quadratic expression is possible if we are able to convert it into a quadratic equation by equalizing it to zero. The values of the variable x which satisfy the quadratic expression and equalize it to zero are called the zeros of the equation. Some of the expressions cannot be easily solved by the method of factorizing. The quadratic formula is here to help. The quadratic formula is also known as "Quadranator." Quadranator alone is enough to solve all quadratic expression problems. The quadratic expressions formula is as follows.

Discriminant

The discriminant is an important part of the quadratic expression formula. The value of the discriminant is (b 2 - 4ac). This is called a discriminant because it discriminates the zeroes of the quadratic expression based on its sign. Many an instance we wish to know more about the zeros of the equation, before calculating the roots of the expression. Here the discriminant value is useful. Based on the values of the discriminant the nature of the zeros of the equation or the roots of the equation can be predicted.

The nature of the roots of the quadratic expression based on the discriminant value is as follows.

• When the quadratic expression has two real and distinct roots: b 2 - 4ac > 0
• When the quadratic expression has equal roots: b 2 - 4ac = 0
• When the quadratic expression does not have any roots or has imaginary roots: b 2 - 4ac < 0

Related Topics

The following topics are related to quadratic expressions.

• Quadratic Equation
• Graphing a Quadratic Function
• Polynomial Functions
• Algebraic Expressions
• Domain and Range of a Parabola

Examples on Quadratic Expressions

Example 1: Jack shows a quadratic expression. The quadratic expression is x 2 + 3x - 4. Can you find the value of expression at x = -2?

Suppose we have an expression p(x) in variable (x).

The value of expression at (k) is given by (p(k)).

To find the value of expression at (x = -2), substitute -2 for (x) in the given expression.

(-2) 2 + 3(-2) -4 = 4- 6 -4 = -6

Therefore the value of expression at x = 2 is -6.

Example 2: Flora loves collecting flowers. She wants to choose a number such that it is the solution to the quadratic expression \(x^{2}-8x+12\). Can you help her choose the correct number?

Compare the quadratic expression \(x^{2}-8x+12\) with the standard form of quadratic expression \(ax^{2}+bx+c\).

We get, \(a=1\), \(b=-8\) and the constant \(c=12\)

Now calculate the solution of the quadratic expression by using the quadratic formula.

\[\begin{aligned}x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\&=\frac{-(-8)\pm\sqrt{(-8)^2-4(1)(12)}}{2}\\&=\frac{8\pm\sqrt{64-48}}{2}\\&=\frac{8\pm\sqrt{16}}{2}\\&=\frac{8\pm4}{2}\\x&=6,2\end{aligned}\]

The number 2 is not on the flowers. Therefore, she will choose the flower on which number 6 is written.

Therefore the correct number is 6.

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Practice Questions on Quadratic Expressions

Faqs on quadratic expressions, what is a quadratic expression in math.

An expression of the form ax 2 + bx + c, where a ≠ 0 is called a quadratic expression in variable x.

What Are the 3 Forms of a Quadratic Expression?

The three forms of quadratic expressions are

• Standard form
• Factored form
• Vertex form

What Does it Mean to Factor a Quadratic Expression?

The factored form of a quadratic expression is to split the quadratic expression into a product of two factors. A quadratic expression is the multiplication of two linear expressions, and the process of factorization is to find the two factors of the quadratic expression..

Which Is a Quadratic Function?

A function of the form f(x) = ax 2 + bx + c, where a ≠ 0 is called a quadratic function in variable x. Similar to a function , a quadratic expression also has a domain and a range value.

What Does Quadratic Mean in a Quadratic Expression?

The word "quadratic" is derived from the word "quad" which means square. In the quadratic expression ax 2 + bx + c, we have the term x 2 which refers to the term quadratic.

How Do you Know If a Polynomial is a Quadratic Expression?

The polynomial expression having a maximum of x 2 term is termed as a quadratic expression. The general form of a quadratic expression is ax 2 + bx + c, and If the coefficient of x 2 is non-zero, then the expression is a quadratic expression.

What Are the Characteristics of a Quadratic Expression?

The characteristics of a quadratic expression are:

• The standard form of the quadratic expression is ax 2 + bx + c, where a ≠ 0.
• A quadratic expression has at most two zeroes.
• The curve of the quadratic expression is in the form of a parabola.

What Are the Terms in a Quadratic Expression?

A quadratic expression generally has three terms. The x 2 term, the x term, and the constant term. The standard form of a quadratic expression is of the form ax 2 + bx + c, and a is a non-zero value. Here a, b are called the coefficients and c is the constant term.

How do you write Quadratic Expressions in Standard Form?

The standard form of a quadratic equation is ax 2 + bx + c, where a ≠ 0 in variable x.

How Do you Simplify a Quadratic Expression?

Quadratic equations can be simplified by the process of factorization. Further, the other methods of solving a quadratic equation are by using the formula, and by the method of finding squares. For a quadratic expression of the form x 2 + (a + b)x + ab, the process of factorization gives the following simplified factors (x + a)(x + b).

• Parallelogram
• Quadrilateral
• Parallelepiped
• Tetrahedron
• Dodecahedron
• Rectangular Prism
• Fibonacci Sequence
• Golden Ratio
• Fraction Calculator
• Mixed Fraction Calculator
• Greatest Common Factor Calulator
• Decimal to Fraction Calculator
• Whole Numbers
• Rational Numbers
• Place Value
• Irrational Numbers
• Natural Numbers
• Binary Operation
• Numerator and Denominator
• Order of Operations (PEMDAS)
• Scientific Notation
• Triangular Number
• Complex Number
• Binary Number System
• Binomial Theorem
• Quartic Function
• Mathematical Induction
• Group Theory
• Modular Arithmetic
• Euler’s Number
• Inequalities
• De Morgan’s Laws
• Transcendental Numbers

Last modified on August 3rd, 2023

Quadratic Equation Word Problems

Here, we will solve different types of quadratic equation-based word problems. Use the appropriate method to solve them:

• By Completing the Square
• By Factoring
• By Quadratic Formula
• By graphing

For each process, follow the following typical steps:

• Make the equation
• Solve for the unknown variable using the appropriate method
• Interpret the result

The product of two consecutive integers is 462. Find the numbers?

Let the numbers be x and x + 1 According to the problem, x(x + 1) = 483 => x 2 + x – 483 = 0 => x 2 + 22x – 21x – 483 = 0 => x(x + 22) – 21(x + 22) = 0 => (x + 22)(x – 21) = 0 => x + 22 = 0 or x – 21 = 0 => x = {-22, 21} Thus, the two consecutive numbers are 21 and 22.

The product of two consecutive positive odd integers is 1 less than four times their sum. What are the two positive integers.

Let the numbers be n and n + 2 According to the problem, => n(n + 2) = 4[n + (n + 2)] – 1 => n 2 + 2n = 4[2n + 2] – 1 => n 2 + 2n = 8n + 7 => n 2 – 6n – 7 = 0 => n 2 -7n + n – 7 = 0 => n(n – 7) + 1(n – 7) = 0 => (n – 7) (n – 1) = 0 => n – 7 = 0 or n – 1 = 0 => n = {7, 1} If n = 7, then n + 2 = 9 If n = 1, then n + 1 = 2 Since 1 and 2 are not possible. The two numbers are 7 and 9

A projectile is launched vertically upwards with an initial velocity of 64 ft/s from a height of 96 feet tower. If height after t seconds is reprented by h(t) = -16t 2 + 64t + 96. Find the maximum height the projectile reaches. Also, find the time it takes to reach the highest point.

Since the graph of the given function is a parabola, it opens downward because the leading coefficient is negative. Thus, to get the maximum height, we have to find the vertex of this parabola. Given the function is in the standard form h(t) = a 2 x + bx + c, the formula to calculate the vertex is: Vertex (h, k) = ${\left\{ \left( \dfrac{-b}{2a}\right) ,h\left( -\dfrac{b}{2a}\right) \right\}}$ => ${\dfrac{-b}{2a}=\dfrac{-64}{2\times \left( -16\right) }}$ = 2 seconds Thus, the time the projectile takes to reach the highest point is 2 seconds ${h\left( \dfrac{-b}{2a}\right)}$ = h(2) = -16(2) 2 – 64(2) + 80 = 144 feet Thus, the maximum height the projectile reaches is 144 feet

The difference between the squares of two consecutive even integers is 68. Find the numbers.

Let the numbers be x and x + 2 According to the problem, (x + 2) 2 – x 2 = 68 => x 2 + 4x + 4 – x 2 = 68 => 4x + 4 = 68 => 4x = 68 – 4 => 4x = 64 => x = 16 Thus the two numbers are 16 and 18

The length of a rectangle is 5 units more than twice the number. The width is 4 unit less than the same number. Given the area of the rectangle is 15 sq. units, find the length and breadth of the rectangle.

Let the number be x Thus, Length = 2x + 5 Breadth = x – 4 According to the problem, (2x + 5)(x – 4) = 15 => 2x 2 – 8x + 5x – 20 – 15 = 0 => 2x 2 – 3x – 35 = 0 => 2x 2 – 10x + 7x – 35 = 0 => 2x(x – 5) + 7(x – 5) = 0 => (x – 5)(2x + 7) = 0 => x – 5 = 0 or 2x + 7 = 0 => x = {5, -7/2} Since we cannot have a negative measurement in mensuration, the number is 5 inches. Now, Length = 2x + 5 = 2(5) + 5 = 15 inches Breadth = x – 4 = 15 – 4 = 11 inches

A rectangular garden is 50 cm long and 34 cm wide, surrounded by a uniform boundary. Find the width of the boundary if the total area is 540 cm².

Given, Length of the garden = 50 cm Width of the garden = 34 cm Let the uniform width of the boundary be = x cm According to the problem, (50 + 2x)(34 + 2x) – 50 × 34 = 540 => 4x 2 + 168x – 540 = 0 => x 2 + 42x – 135 = 0 Since, this quadratic equation is in the standard form ax 2 + bx + c, we will use the quadratic formula, here a = 1, b = 42, c = -135 x = ${x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}}$ => ${\dfrac{-42\pm \sqrt{\left( 42\right) ^{2}-4\times 1\times \left( -135\right) }}{2\times 1}}$ => ${\dfrac{-42\pm \sqrt{1764+540}}{2}}$ => ${\dfrac{-42\pm \sqrt{2304}}{2}}$ => ${\dfrac{-42\pm 48}{2}}$ => ${\dfrac{-42+48}{2}}$ and ${\dfrac{-42-48}{2}}$ => x = {-45, 3} Since we cannot have a negative measurement in mensuration the width of the boundary is 3 cm

The hypotenuse of a right-angled triangle is 20 cm. The difference between its other two sides is 4 cm. Find the length of the sides.

Let the length of the other two sides be x and x + 4 According to the problem, (x + 4) 2 + x 2 = 20 2 => x 2 + 8x + 16 + x 2 = 400 => 2x 2 + 8x + 16 = 400 => 2x 2 + 8x – 384 = 0 => x 2 + 4x – 192 = 0 => x 2 + 16x – 12x – 192 = 0 => x(x + 16) – 12(x + 16) = 0 => (x + 16)(x – 12) = 0 => x + 16 = 0 and x – 12 = 0 => x = {-16, 12} Since we cannot have a negative measurement in mensuration, the lengths of the sides are 12 and 16

Jennifer jumped off a cliff into the swimming pool. The function h can express her height as a function of time (t) = -16t 2 +16t + 480, where t is the time in seconds and h is the height in feet. a) How long did it take for Jennifer to attain a maximum length. b) What was the highest point that Jennifer reached. c) Calculate the time when Jennifer hit the water?

Comparing the given function with the given function f(x) = ax 2 + bx + c, here a = -16, b = 16, c = 480 a) Finding the vertex will give us the time taken by Jennifer to reach her maximum height  x = ${-\dfrac{b}{2a}}$ = ${\dfrac{-16}{2\left( -16\right) }}$ = 0.5 seconds Thus Jennifer took 0.5 seconds to reach her maximum height b) Putting the value of the vertex by substitution in the function, we get ${h\left( \dfrac{1}{2}\right) =-16\left( \dfrac{1}{2}\right) ^{2}+16\left( \dfrac{1}{2}\right) +480}$ => ${-16\left( \dfrac{1}{4}\right) +8+480}$ => 484 feet Thus the highest point that Jennifer reached was 484 feet c) When Jennifer hit the water, her height was 0 Thus, by substituting the value of the height in the function, we get -16t 2 +16t + 480 = 0 => -16(t 2 + t – 30) = 0 => t 2 + t – 30 = 0 => t 2 + 6t – 5t – 30 = 0 => t(t + 6) – 5(t + 6) = 0 => (t + 6)(t – 5) = 0 => t + 6 = 0 or t – 5 = 0 => x = {-6, 5} Since time cannot have any negative value, the time taken by Jennifer to hit the water is 5 seconds.

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