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How to Solve Quadratic Equations

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Quadratic equations are equations with the form

a x^{2} + b x + c = 0 .

The expression $a x^{2} + b x + c$ is called a quadratic expression, because the highest power of any of the terms is 2. There are four methods for solving quadratic equations by hand:

1.: The quadratic formula
2.: Solving quadratic equations where $c = 0$
3.: Solving quadratic equations where $b = 0$
4.: Solving quadratic equations by inspection

The results of these methods will always be the roots of the function.

Formula

The Quadratic Formula

The quadratic formula can be used with all quadratic expressions. The roots are

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} .

Quadratic equations can have no solutions, one solution or two solutions.

$b^{2} - 4 a c < 0 \Rightarrow$ no real solutions,
$b^{2} - 4 a c = 0 \Rightarrow$ one real solution,
$b^{2} - 4 a c > 0 \Rightarrow$ two real solutions.

Example 1

Solve the equation $x^{2} + 11 x = - 30$ .

First, you have to put all the nonzero terms on one side of the equal sign to get $0$ alone on the other side:

\begin{array}{l} x^{2} + 11 x & = - 30 \\ x^{2} + 11 x + 30 & = 0 \end{array}

Next, you use the quadratic formula with $a = 1$ , $b = 11$ and $c = 30$ : $\begin{array}{l} x & = \frac{- 11 \pm \sqrt{{(11)}^{2} - 4 \cdot 1 \cdot 30}}{2 \cdot 1} \\ = \frac{- 11 \pm \sqrt{121 - 120}}{2} \\ = \frac{- 11 \pm \sqrt{1}}{2} \\ = \frac{- 11 \pm 1}{2} \end{array}$

Set up the expressions with both the positive square root and the negative square root:

\begin{array}{l} x_{1} & = \frac{- 11 + 1}{2} & x_{2} & = \frac{- 11 - 1}{2} \\ = \frac{- 10}{2} & = \frac{- 12}{2} \\ = - 5 & = - 6 \end{array}

That means the solutions are $x_{1} = - 5$ and $x_{2} = - 6$ .

Rule

Solving a Quadratic Equation Where $c = 0$

When $c = 0$ , the expression looks like this:

a x^{2} + b x = 0

1.: Factorize the expression by putting an x outside the parentheses, which gives you a product in the form $p \cdot q = 0$ .
2.: That means you can do this: $p \cdot q = 0 \Leftrightarrow p = 0$ or $q = 0$ .

Example 2

Solve the equation $2 x^{2} + 4 x = 0$ .

You factorize by taking $2 x$ out of the parentheses:

2 x (x + 2) = 0 .

Then you set up an equation for each factor:

\begin{array}{l} 2 x & = 0 & x + 2 & = 0 \\ x & = 0 & x & = - 2 \end{array}

That will give you $x_{1} = 0$ and $x_{2} = - 2$ .

Rule

Solving Quadratic Equations Where $b = 0$

When $b = 0$ , the expression looks like this:

a x^{2} + c = 0

1.: Move the constant $c$ over to the other side.
2.: Divide by $a$ .
3.: Take the square root of both sides.
4.: You have a positive and a negative solution.

Example 3

\begin{array}{l} 3 x^{2} - 27 & = 0 \\ 3 x^{2} & = 27 | \div 3 \\ x^{2} & = 9 \\ x & = \pm 3 \end{array}

Rule

Solving a Quadratic Equation by Inspection

You have $a x^{2} + b x + c = 0$ . When you solve by inspection, you follow these two rules:

\begin{array}{l} c & = p \cdot q, \\ b & = p + q . \end{array}

Here, $x_{1} = - p$ and $x_{2} = - q$ are the roots of the function, making them the solutions to the equation.

Example 4

Solve the equation $x^{2} - x - 56 = 0$ .

You can see that $b = - 1$ and $c = - 56$ . You need to find values for $p$ and $q$ that can get you the solution to the equation. There are multiple combinations of factors that give you the product $- 56$ . Here are some of them:

\begin{array}{l} - (2 \cdot 28) & = - 56, \\ - (4 \cdot 14) & = - 56, \\ - (7 \cdot 8) & = - 56, \end{array}

As all of these products are $- 56$ , they are all candidates for the solution. For each of these products, we want a negative difference between the factors, because you are looking for the answer $- 1$ , a negative number. That means we set the differences up like this:

\begin{array}{l} 2 - 28 & = - 26 \neq - 1 \\ 4 - 14 & = - 10 \neq - 1 \\ 7 - 8 & = - 1, \end{array}

That shows you that $p = 7$ and $q = - 8$ fits with the equation. According to the formula, that means

x_{1} = - 7 and x_{2} = - (- 8) = 8,

because you have to change the sign to find the solution.

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Go back

The Quadratic Formula

Solving a Quadratic Equation Where c = 0

Solving Quadratic Equations Where b = 0

Solving a Quadratic Equation by Inspection

Solving a Quadratic Equation Where $c = 0$

Solving Quadratic Equations Where $b = 0$