Solve ) 5m ^ 2 = 22m + 15 | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$5 m ^ { 2 } = 22 m + 15$$

Solve for m

$m=-\frac{3}{5}=-0.6$
$m=5$

Steps Using Factoring By Grouping

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using Factoring By Grouping

Subtract $22m$ from both sides.

$$5m^{2}-22m=15$$

Subtract $15$ from both sides.

$$5m^{2}-22m-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as $5m^{2}+am+bm-15$. To find $a$ and $b$, set up a system to be solved.

$$a+b=-22$$ $$ab=5\left(-15\right)=-75$$

Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product $-75$.

$$1,-75$$ $$3,-25$$ $$5,-15$$

Calculate the sum for each pair.

$$1-75=-74$$ $$3-25=-22$$ $$5-15=-10$$

The solution is the pair that gives sum $-22$.

$$a=-25$$ $$b=3$$

Rewrite $5m^{2}-22m-15$ as $\left(5m^{2}-25m\right)+\left(3m-15\right)$.

$$\left(5m^{2}-25m\right)+\left(3m-15\right)$$

Factor out $5m$ in the first and $3$ in the second group.

$$5m\left(m-5\right)+3\left(m-5\right)$$

Factor out common term $m-5$ by using distributive property.

$$\left(m-5\right)\left(5m+3\right)$$

To find equation solutions, solve $m-5=0$ and $5m+3=0$.

$$m=5$$ $$m=-\frac{3}{5}$$

Steps Using the Quadratic Formula

Subtract $22m$ from both sides.

$$5m^{2}-22m=15$$

Subtract $15$ from both sides.

$$5m^{2}-22m-15=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $5$ for $a$, $-22$ for $b$, and $-15$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$m=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 5\left(-15\right)}}{2\times 5}$$

Square $-22$.

$$m=\frac{-\left(-22\right)±\sqrt{484-4\times 5\left(-15\right)}}{2\times 5}$$

Multiply $-4$ times $5$.

$$m=\frac{-\left(-22\right)±\sqrt{484-20\left(-15\right)}}{2\times 5}$$

Multiply $-20$ times $-15$.

$$m=\frac{-\left(-22\right)±\sqrt{484+300}}{2\times 5}$$

Add $484$ to $300$.

$$m=\frac{-\left(-22\right)±\sqrt{784}}{2\times 5}$$

Take the square root of $784$.

$$m=\frac{-\left(-22\right)±28}{2\times 5}$$

The opposite of $-22$ is $22$.

$$m=\frac{22±28}{2\times 5}$$

Multiply $2$ times $5$.

$$m=\frac{22±28}{10}$$

Now solve the equation $m=\frac{22±28}{10}$ when $±$ is plus. Add $22$ to $28$.

$$m=\frac{50}{10}$$

Divide $50$ by $10$.

$$m=5$$

Now solve the equation $m=\frac{22±28}{10}$ when $±$ is minus. Subtract $28$ from $22$.

$$m=-\frac{6}{10}$$

Reduce the fraction $\frac{-6}{10}$ to lowest terms by extracting and canceling out $2$.

$$m=-\frac{3}{5}$$

The equation is now solved.

$$m=5$$ $$m=-\frac{3}{5}$$

Steps for Completing the Square

Subtract $22m$ from both sides.

$$5m^{2}-22m=15$$

Divide both sides by $5$.

$$\frac{5m^{2}-22m}{5}=\frac{15}{5}$$

Dividing by $5$ undoes the multiplication by $5$.

$$m^{2}-\frac{22}{5}m=\frac{15}{5}$$

Divide $15$ by $5$.

$$m^{2}-\frac{22}{5}m=3$$

Divide $-\frac{22}{5}$, the coefficient of the $x$ term, by $2$ to get $-\frac{11}{5}$. Then add the square of $-\frac{11}{5}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{22}{5}m+\left(-\frac{11}{5}\right)^{2}=3+\left(-\frac{11}{5}\right)^{2}$$

Square $-\frac{11}{5}$ by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{22}{5}m+\frac{121}{25}=3+\frac{121}{25}$$

Add $3$ to $\frac{121}{25}$.

$$m^{2}-\frac{22}{5}m+\frac{121}{25}=\frac{196}{25}$$

Factor $m^{2}-\frac{22}{5}m+\frac{121}{25}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(m-\frac{11}{5}\right)^{2}=\frac{196}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{11}{5}\right)^{2}}=\sqrt{\frac{196}{25}}$$

Simplify.

$$m-\frac{11}{5}=\frac{14}{5}$$ $$m-\frac{11}{5}=-\frac{14}{5}$$

Add $\frac{11}{5}$ to both sides of the equation.

$$m=5$$ $$m=-\frac{3}{5}$$