Section 1.5 - Quadratic Equations

Section Objectives

1. Solve quadratic equations by factoring.
2. Solve quadratic equations by using square roots.
3. Solve quadratic equations by using the quadratic formula.

A quadratic equation in the variable $x$ is an equation of the form $ax^2+bx+c=0$, where $a \ne 0$. This is an example of a polynomial equation. In fact, it is a 2nd degree polynomial equation.

There are three common approaches to solving quadratic equations:

1. Factor and set each factor to zero.
2. Use square roots to "undo" the square (after completing the square).
3. Use the quadratic formula.

Using Factoring to Solve Quadratic Equations

If several numbers (or expressions) are multiplied and their product is zero, then one of the factors must be zero. We normally state this property by saying:

If $AB=0$, then $A=0$ or $B=0$.

This property allows us to solve quadratic (or higher-order) equations by factoring. For a quick summary of factoring techniques, see this sheet.

Examples

• [1] Solve for $x$: $\quad (2x+1)(x-5)=0$ (Solution)

• [2] Solve for $x$: $\quad 3x(x-1)(x+3)(x-9)=0$ (Solution)

• [3] Solve for $x$ by factoring: $\quad x^2-2x-3=0$ (Solution)

• [4] Solve for $v$ by factoring: $\quad v^2+v-2=0$ (Solution)

• [5] Solve for $u$ by factoring: $\quad 5u^2-6=-13u$ (Solution)

• [6] Solve for $w$ by factoring: $\quad 4w^2+7w=-3$ (Solution)

Using Square Roots to Solve Quadratic Equations

A quadratic equation that has the form $X^2=a$ can be solved by taking square roots of both sides of the equation. Remember that a perfect square usually has two square roots, one positive and one negative.

If $X^2=a$, then $X=\sqrt{a}$ or $X = -\sqrt{a}$.

Examples

• [7] Solve for $x$ by using square roots: $\quad x^2+3=12$ (Solution)

• [8] Solve for $w$ by using square roots: $\quad (w-6)^2=49$ (Solution)

• [9] Solve for $x$ by using square roots: $\quad (2x+3)^2 = 0$ (Solution)

Completing the Square

We can always use square roots to solve a quadratic equation as long as the square "is complete." To complete the square means to rewrite a standard form quadratic expression in terms of a complete square:

$ax^2+bx+c = a(x-h)^2+k$

With practice, completing the square is usually pretty straight-forward. This sheet walks you through the process step by step. Refer to the sheet if you are having trouble. But it might help to just use the formulas:

Examples

• [10] Complete the square: $\quad x^2+2x$ (Solution)

• [11] Solve by completing the square: $\quad x^2+4x+3=0$ (Solution)

• [12] Solve by completing the square: $\quad x^2+13=6x$ (Solution)

Quadratic Formula

By completing the square, we can derive a formula that is capable giving the solutions to any quadratic equation. This formula is called the quadratic formula:

If $ax^2+bx+c=0$, then $\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

The quantity under the radical, $b^2-4ac$, is called the discriminant. Its sign tells us about the solutions:

1. If $b^2-4ac=0$, then there is only one real number solution.
2. If $b^2-4ac>0$, then there are two different real number solutions.
3. If $b^2-4ac<0$, then there are two different complex number solutions.

Examples

• [13] Solve for $x$: $\quad 3x^2-8x+1=0$ (Solution)

• [14] Solve for $x$: $\quad 4x^2+x-4=0$. Round your answer(s) to the nearest hundredth. (Solution)

• [15] Solve for $x$: $\quad 3x^2+3x+7=0$ (Solution)

• [16] Use the discriminant to determine the nature of the solutions: $\quad -4x^2+7x-1=0$ (Solution)

• [17] A ball is thrown from a height of 67 meters with an initial downward velocity of 5 meters per second. The height of the ball (in meters) after $t$ seconds is given by $h=67-5t-4.9t^2$. When does the ball hit the ground? Round your answer to the nearest hundredth. (Solution)

For further help...

1. Khan Academy has some excellent videos on quadratic equations and applications. Start at this page.
2. Do a Google search for "solving quadratic equations" or "applications of quadratic equations."