Solve Find the mean, median and mode of the following data. a. 211036423952 30 2526 overline x =; tilde x =\; hat x =\ b . 215525 30 263642393625 overline x =; tilde x =; hat x =\ C. 108 120 154 118 125 164 135 overline x =\; tilde x =\; hat x =\ d. 31 21 16 15212719 18 overline x = tilde x =; hat x = e. 87 94 36 56 54 76 87 856856 78 88 overline x = overline x = hat x =

Question

$$\left. \begin{array} { l } { x = 4 x x = 8 } \\ { x = 9 x = x x = 8 } \\ { x = x = 4 \text { a n d \right.$$

Solve for x

$x=\frac{1}{4}=0.25$
$x=0$

Steps Using Factoring

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using Factoring

Multiply $x$ and $x$ to get $x^{2}$.

$$x=4x^{2}$$

Subtract $4x^{2}$ from both sides.

$$x-4x^{2}=0$$

Factor out $x$.

$$x\left(1-4x\right)=0$$

To find equation solutions, solve $x=0$ and $1-4x=0$.

$$x=0$$ $$x=\frac{1}{4}$$

Steps Using the Quadratic Formula

Multiply $x$ and $x$ to get $x^{2}$.

$$x=4x^{2}$$

Subtract $4x^{2}$ from both sides.

$$x-4x^{2}=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.

$$-4x^{2}+x=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-4$ for $a$, $1$ for $b$, and $0$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

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

Take the square root of $1^{2}$.

$$x=\frac{-1±1}{2\left(-4\right)}$$

Multiply $2$ times $-4$.

$$x=\frac{-1±1}{-8}$$

Now solve the equation $x=\frac{-1±1}{-8}$ when $±$ is plus. Add $-1$ to $1$.

$$x=\frac{0}{-8}$$

Divide $0$ by $-8$.

$$x=0$$

Now solve the equation $x=\frac{-1±1}{-8}$ when $±$ is minus. Subtract $1$ from $-1$.

$$x=-\frac{2}{-8}$$

Reduce the fraction $\frac{-2}{-8}$ to lowest terms by extracting and canceling out $2$.

$$x=\frac{1}{4}$$

The equation is now solved.

$$x=0$$ $$x=\frac{1}{4}$$

Steps for Completing the Square

Multiply $x$ and $x$ to get $x^{2}$.

$$x=4x^{2}$$

Subtract $4x^{2}$ from both sides.

$$x-4x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.

$$-4x^{2}+x=0$$

Divide both sides by $-4$.

$$\frac{-4x^{2}+x}{-4}=\frac{0}{-4}$$

Dividing by $-4$ undoes the multiplication by $-4$.

$$x^{2}+\frac{1}{-4}x=\frac{0}{-4}$$

Divide $1$ by $-4$.

$$x^{2}-\frac{1}{4}x=\frac{0}{-4}$$

Divide $0$ by $-4$.

$$x^{2}-\frac{1}{4}x=0$$

Divide $-\frac{1}{4}$, the coefficient of the $x$ term, by $2$ to get $-\frac{1}{8}$. Then add the square of $-\frac{1}{8}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{4}x+\left(-\frac{1}{8}\right)^{2}=\left(-\frac{1}{8}\right)^{2}$$

Square $-\frac{1}{8}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{1}{64}$$

Factor $x^{2}-\frac{1}{4}x+\frac{1}{64}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{1}{8}\right)^{2}=\frac{1}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{8}\right)^{2}}=\sqrt{\frac{1}{64}}$$

Simplify.

$$x-\frac{1}{8}=\frac{1}{8}$$ $$x-\frac{1}{8}=-\frac{1}{8}$$

Add $\frac{1}{8}$ to both sides of the equation.

$$x=\frac{1}{4}$$ $$x=0$$

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Example

Question

Solve for x

Steps Using Factoring

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using Factoring

Steps Using the Quadratic Formula

Steps for Completing the Square