Quadratic Equation Questions

Introduction

A quadratic equation is a polynomial equation with the highest degree of two. The values satisfying the quadratic equation are called roots of them. There are a few methods to solve a quadratic equation and find its roots. The roots can be calculated using the factorization method by splitting the middle term, converting the quadratic equation into a complete square, and using the quadratic formula.

Quadratic Equations

A Quadratic equation is a polynomial equation of one variable with a degree of two. The general form of a quadratic equation f(x)=ax²+bx+c=0, in which x is the unknown variable, a≠0, and a,b,c ϵ R. a is the leading coefficient of the quadratic equation, c is the absolute term of the quadratic equation. Example: 3x²+5x+6=0, -x²+2x-1=0, etc…

Methods to Solve Quadratic Equations

The values of the unknown variable that on substituting in the quadratic equation results in zero are called the roots of the quadratic equation. It has two roots because the degree of a quadratic equation is equal to two.

The following methods are used to calculate the roots or solve a quadratic equation −

Factorization by Splitting the middle term

Consider the quadratic equation f(x)=ax^2+bx+c=0. Now to find the roots of the quadratic equation using factorization by splitting the middle term method let’s take two numbers p,q such that the product of the numbers is equal to the product of a and c, and the sum of the numbers is equal to b.

p×q=a×c and p+q=b

Now substitute p+q=b

$$\mathrm{\mathit{f}(x)=ax^2+(p+q)x+c=0}$$

$$\mathrm{\mathit{f}(x)=ax^2+px+qx+c=0}$$

Now using p×q=a×c take the common factors and write the equation as the product of two factors, and then equate each factor to zero resulting in two values of x which are the roots.

Examples of Splitting the middle term

1) Solve the quadratic equation x²+7x+12=0 using the factorization by splitting the middle term?

The values of a,b,c are 1, 7, 12 respectively.

Let’s take the product of a,c that is 12. Now, write down the factors of 12.

Factors of 12 = 1, 2, 3, 4, 6, 12

Now, search for two factors whose product is equal to 12 and the sum is equal to 7.

3, 4 satisfy the condition.

Now, split the middle term using the sum, in the quadratic equation

$$\mathrm{x^2+7x+12=0}$$

$$\mathrm{x^2+3x+4x+12=0}$$

Now, take the common factors

$$\mathrm{x(x+3)+4(x+3)=0}$$

Write down the above equation as the product of two factors by taking the common factor out once again.

$$\mathrm{(x+3)(x+4)=0}$$

These two are the factors of the quadratic equation solve the linear equations separately to get the roots of the quadratic equation.

$$\mathrm{x+3=0; x+4=0}$$

x=-3,-4 are the roots.

2)Solve the quadratic equation x^2-5x+6=0 using the factorization by splitting the middle term?

The values of a,b,c are 1, -5, 6 respectively.

Let’s take the product of a,c that is 6. Now, write down the factors of 6.

Factors of 6 = 1, 2, 3, 6

Now, search for two factors whose product is equal to 6 and the sum is equal to 5.

-2, -3 satisfy the condition.

Now, split the middle term using the sum, in the quadratic equation

$$\mathrm{x^2-5x+6=0}$$

$$\mathrm{x^2-2x-3x+6=0}$$

Now, take the common factors

$$\mathrm{x(x-2)-3(x-2)=0}$$

Write down the above equation as the product of two factors by taking the common factor out once again.

$$\mathrm{(x-2)(x-3)=0}$$

These two are the factors of the quadratic equation solve the linear equations separately to get the roots of the quadratic equation.

$$\mathrm{x-2=0; x-3=0}$$

x=2,3 are the roots.

Completing the Square

Consider the quadratic equation f(x)=ax²+bx+c=0, in which x is the unknown variable, a≠0, and a,b,c ϵ R. Now to find the roots of the quadratic equation using completing the square method send c to the other side of the equation.

$$\mathrm{ax^2+bx=-c}$$

Now make sure the coefficient of x^2 is 1. Divide both sides of the equation with $\mathrm{\frac{1}{a}}$, if a≠1.

$$\mathrm{x^2+\frac{b}{a} x=-\frac{c}{a}}$$

Now, add $\mathrm{(\frac{b}{2a})^2 }$ to both sides of the equation to make a perfect square on the left-hand side or $\mathrm{(\frac{b}{2})^2 }$ if the value of a=1

$$\mathrm{x^2+\frac{b}{a}x+ (\frac{b}{2a} )^2=-\frac{c}{a}+ (\frac{b}{2a})^2}$$

$$\mathrm{(x+\frac{b}{2a})^2=-\frac{c}{a}+ (\frac{b}{2a})^2}$$

Now, taking the square root on both sides of the equation and solving gives the roots of the quadratic equation.

Examples of Completing the Square

1)Solve the quadratic equation x²+6x-7=0 using the completing the square method?

Send the constant term to the other side of the equation,

$$\mathrm{x^2+6x=7}$$

The coefficient of x² is 1, so add $\mathrm{(\frac{b}{2})^2}$ to both sides of the equation to make a perfect square on the left-hand side.

Adding (3)² on both sides of the equation,

$$\mathrm{x^2+6x+9=7+9=16}$$

$$\mathrm{(x+3)^2=16}$$

Now, taking square root on both sides gives,

$$\mathrm{x+3=±4}$$

x+3=4 and x+3=-4

x=1,-7 are the roots.

2)Solve the quadratic equation 2x²+5x+3=0 using the completing the square method?

Send the constant term to the other side of the equation,

$$\mathrm{2x^2+5x=-3}$$

The coefficient of x² is 2, so divide the equation by $\mathrm{\frac{1}{2}}$.

$$\mathrm{x^2+\frac{5}{2} x=-\frac{3}{2}}$$

add $\mathrm{(\frac{b}{2a})^2}$ to both sides of the equation to make a perfect square on the left-hand side.

Adding $\mathrm{(\frac{5}{4})^2}$ on both sides of the equation,

$$\mathrm{x^2+\frac{5}{2} x+(\frac{5}{4})^2=-\frac{3}{2}+(\frac{5}{4})^2}$$

Now, taking square root on both sides gives,

$$\mathrm{(x+\frac{5}{4})^2=\frac{1}{16}}$$

$$\mathrm{x+\frac{5}{4}=±\frac{1}{4}}$$

x=-1,$\mathrm{-\frac{3}{2}}$ are the roots.

Quadratic Formula

Consider the quadratic equation f(x)=ax²+bx+c=0, in which x is the unknown variable, a≠0, and a,b,c ϵ R. Now, the roots of the quadratic equation using the quadratic formula are −

$\mathrm{x=\frac{-b±\sqrt{b^2-4ac}}{2a}}$, substitute the respective values of a,b,c in the formula.

Examples of Quadratic Formula

1) Solve the quadratic equation x^2+4x+1=0 using the quadratic formula?

The values of a,b,c are 1, 4, 1 respectively.

$\mathrm{x=\frac{-b±\sqrt{b^2-4ac}}{2a}}$, substitute the respective values of a,b,c in the formula to get the roots of the equation.

$$\mathrm{x=\frac{-4±\sqrt{4^2-4}}{2}=\frac{-4±√12}{2}=-2±√3}$$

-2+√3,-2-√3 are the roots.

2) Solve the quadratic equation x²+3x+6=0 using the quadratic formula?

The values of a,b,c are 1, 3, 6 respectively.

$\mathrm{x=\frac{-b±\sqrt{b^2-4ac}}{2a}}$, substitute the respective values of a,b,c in the formula to get the roots of the equation.

$$\mathrm{x=\frac{-3±\sqrt{3^2-24}}{2}=\frac{-3±\sqrt{-15}}{2}=\frac{-3±i\sqrt{15}}{2}}$$

$\mathrm{\frac{-3+i\sqrt{15}}{2},\frac{-3-i\sqrt{15}}{2}}$are the roots.

Conclusion

In this tutorial, we learned about quadratic equations, methods to solve them, the method of factorization by splitting the middle term, its examples, the method of completing the square, its examples, quadratic formula, and its examples.

FAQs

1.What is the value of the leading coefficient in a quadratic equation 2x²+3x+4=0?

The value of the leading coefficient is 2.

2.What is the quadratic formula for the quadratic equation ax²+bx+c=0?

$\mathrm{x=\frac{-b±\sqrt{b^2-4ac}}{2a}}$, substitute the respective values of a,b,c in the formula.

3.What are the methods to solve a quadratic equation?

• Factorization by Splitting the middle term

• Completing the Square

• Quadratic Formula

4.What is the value of the absolute term in a quadratic equation x²-2x+5=0?

The value of the absolute term is equal to 5.

5.What if the coefficient of x² in a polynomial equation is equal to zero?

Then the polynomial equation is not quadratic.