# Algebra 1 - Quadratic Equations

## Introduction

• Quadratic Equation in One Variable is a mathematical sentence of degree 2 that can be written in the standard form .
• The members of the equation are $a{x}^{2}=$quadratic term, $bx=$linear term, and $c=$constant.
• The equation $3{x}^{2}+5x-2=0$ is a complete quadratic equation with .
• The equation $2x\left(x-9\right)=0$ is also a quadratic form but it is not simplified in standard form.
• In the case of $\left(4x-3\right)\left(2x+1\right)=0$, the quadratic equation is expressed in factored form. You need to use the FOIL method to write this in standard form.

## Solved Examples (Extraction of Square Roots)

• Finding the roots or solutions of a quadratic equation implies finding all permissible values of $x$ in $a{x}^{2}+bx+c=0$.
• Quadratic equations in the form ${x}^{2}=b$ can be solved by applying the following concepts:
• If $b>0$, then ${x}^{2}=b$ has two real solutions or roots, $x=±\sqrt{b}$.
• If $b=0$, then ${x}^{2}=b$ has one real solution or root, $x=0$.
• If $b<0$, then ${x}^{2}=b$ has no real solutions or roots.
• The principle in these concepts is facilitated using the Square Root Property: If , then $x=±\sqrt{b}$.

Example 1. What are the roots of $3{\left(x-5\right)}^{2}=135?$

Solution:

$3{\left(x-5\right)}^{2}-135=0\to 3{\left(x-5\right)}^{2}=135$

$\frac{3{\left(x-5\right)}^{2}}{3}=\frac{135}{3}\to {\left(x-5\right)}^{2}=45$

$\sqrt{{\left(x-5\right)}^{2}}=\sqrt{45}$

$x-5=±3\sqrt{5}$

## Solved Examples (Factoring and the Po-Shen Loh Method)

• Apply the different ways of solving a quadratic trinomial of .

Example 1. Solve for the roots of ${a}^{2}-11a-26=0$.

Solution:

Using factoring, we write the trinomial on the left side as $\left(a-13\right)\left(a+2\right)=0$.

Using the Zero Product Property, we have:

$\left(a-13\right)\left(a+2\right)=0$

Example 2. Find the roots of $3{x}^{2}+11x-20=0$.

Solution:

In cases of , we use the ac test method.

To illustrate, we have:

$3{x}^{2}+11x-20=0$

$ac=\left(3\right)\left(-20\right)=-60$

Think of factors of $-60$ whose sum is $11$. The two factors are .

Using the factors of $-60$, we express the linear term of the equation as .

$3{x}^{2}+15x-4x-20=0$

Take the common factors as $3x\left(x+5\right)-4\left(x+5\right)=0$.

$3x\left(x+5\right)-4\left(x+5\right)=0\to \left(x+5\right)\left(3x-4\right)=0$

Thus, the roots are

Solving quadratic equations using Po-Shen Loh Method:

Example 1. What are the roots of $5{x}^{2}+18x-8=0$?

Solution:

For cases like , we divide each term of both sides of the equation by $a$.

$\frac{5{x}^{2}}{5}+\frac{18x}{5}-\frac{8}{5}=\frac{0}{5}$

${x}^{2}+\frac{18x}{5}-\frac{8}{5}=0$

The two numbers whose sum is $-\frac{18}{5}$ are

The two roots of the equation is written as

Using the idea of product of $-\frac{8}{5}$, we have $\left(-\frac{9}{5}-u\right)\left(-\frac{9}{5}+u\right)=-\frac{8}{5}$

Solving for $u$ yields $±\frac{11}{5}$. Use either

${x}_{1}=-\frac{9}{5}-\frac{11}{5}=-4$

${x}_{2}=-\frac{9}{5}+\frac{11}{5}=\frac{2}{5}$

## Solved Examples (Completing the Square and Quadratic Formula)

Example 1. Solve the equation ${x}^{2}-4x-96=0$ using completing the square or quadratic formula.

Solution 1. Using Completing the Square

Rewrite the equation as ${x}^{2}-4x=96$.

Divide the coefficient of the linear term by 2 and square the result. The number obtained must be added to both sides of the equation.

$\frac{4}{2}=2\to {\left(2\right)}^{2}=4$

${x}^{2}-4x+4=96+4$

${\left(x-2\right)}^{2}=100$

$\sqrt{{\left(x-2\right)}^{2}}=\sqrt{100}$

$x-2=±10$

Thus, the solutions are

Solution 2. Using the Quadratic Formula

Use the formula $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$. By direct substitution of , we have:

$x=\frac{4±\sqrt{{\left(-4\right)}^{2}-4\left(1\right)\left(-96\right)}}{2}\to x=\frac{4±20}{2}$

The roots are

## Cheat Sheet

• Use the discriminant $D={b}^{2}-4ac$ to determine the nature of the roots of a quadratic equation.
• If ${b}^{2}-4ac>0$ and not a perfect square, then there are two real, irrational roots.
• If ${b}^{2}-4ac>0$ and a perfect square, then there are two unequal real rational roots.
• If ${b}^{2}-4ac=0$, then there is only one real and rational root.
• if ${b}^{2}-4ac<0$, there are no real roots.
• To formulate a quadratic equation with the given roots , use the model $x-\left({x}_{1}+{x}_{2}\right)x+\left({x}_{1}\right)\left({x}_{2}\right)=0$.

## Blunder Areas

• Completing the Square, Quadratic Formula, and Po-Shen Loh Method are applicable for quadratic equations having trinomials either factorable or non-factorable.
• Always be mindful of the signs when solving for the roots of the quadratic equation.
• When ${x}^{2}+3x+2=0$ is written as $\left(x+2\right)\left(x+1\right)=0$, it is a common mistake to write the roots as . The roots must be .
• If you are asked to write the solution or roots of a quadratic equation, then use the term "or" instead of "and." For instance, we have: . It is incorrect to write it as .
• If you are asked to write the solution set of a quadratic equation, then write it as .
• Be careful in dealing with signs of the coefficients of $a{x}^{2}+bx+c=0$ when using the quadratic formula.