Solve (3x - 1)(72 * 7) - 70 = (6x + 2) ^ 2 | Equatix - Math Solver

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Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$(3x-1)(72\times+7)-70=(6x+2)^{2}$$

Solve for x

$x=\frac{7\sqrt{302}}{6}+\frac{62}{3}\approx 40.94117173$
$x=-\frac{7\sqrt{302}}{6}+\frac{62}{3}\approx 0.392161604$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Multiply $72$ and $7$ to get $504$.

$$\left(3x-1\right)\times 504-70=\left(6x+2\right)^{2}$$

Use the distributive property to multiply $3x-1$ by $504$.

$$1512x-504-70=\left(6x+2\right)^{2}$$

Subtract $70$ from $-504$ to get $-574$.

$$1512x-574=\left(6x+2\right)^{2}$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(6x+2\right)^{2}$.

$$1512x-574=36x^{2}+24x+4$$

Subtract $36x^{2}$ from both sides.

$$1512x-574-36x^{2}=24x+4$$

Subtract $24x$ from both sides.

$$1512x-574-36x^{2}-24x=4$$

Combine $1512x$ and $-24x$ to get $1488x$.

$$1488x-574-36x^{2}=4$$

Subtract $4$ from both sides.

$$1488x-574-36x^{2}-4=0$$

Subtract $4$ from $-574$ to get $-578$.

$$1488x-578-36x^{2}=0$$

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.

$$-36x^{2}+1488x-578=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-36$ for $a$, $1488$ for $b$, and $-578$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-1488±\sqrt{1488^{2}-4\left(-36\right)\left(-578\right)}}{2\left(-36\right)}$$

Square $1488$.

$$x=\frac{-1488±\sqrt{2214144-4\left(-36\right)\left(-578\right)}}{2\left(-36\right)}$$

Multiply $-4$ times $-36$.

$$x=\frac{-1488±\sqrt{2214144+144\left(-578\right)}}{2\left(-36\right)}$$

Multiply $144$ times $-578$.

$$x=\frac{-1488±\sqrt{2214144-83232}}{2\left(-36\right)}$$

Add $2214144$ to $-83232$.

$$x=\frac{-1488±\sqrt{2130912}}{2\left(-36\right)}$$

Take the square root of $2130912$.

$$x=\frac{-1488±84\sqrt{302}}{2\left(-36\right)}$$

Multiply $2$ times $-36$.

$$x=\frac{-1488±84\sqrt{302}}{-72}$$

Now solve the equation $x=\frac{-1488±84\sqrt{302}}{-72}$ when $±$ is plus. Add $-1488$ to $84\sqrt{302}$.

$$x=\frac{84\sqrt{302}-1488}{-72}$$

Divide $-1488+84\sqrt{302}$ by $-72$.

$$x=-\frac{7\sqrt{302}}{6}+\frac{62}{3}$$

Now solve the equation $x=\frac{-1488±84\sqrt{302}}{-72}$ when $±$ is minus. Subtract $84\sqrt{302}$ from $-1488$.

$$x=\frac{-84\sqrt{302}-1488}{-72}$$

Divide $-1488-84\sqrt{302}$ by $-72$.

$$x=\frac{7\sqrt{302}}{6}+\frac{62}{3}$$

The equation is now solved.

$$x=-\frac{7\sqrt{302}}{6}+\frac{62}{3}$$ $$x=\frac{7\sqrt{302}}{6}+\frac{62}{3}$$

Steps for Completing the Square

Multiply $72$ and $7$ to get $504$.

$$\left(3x-1\right)\times 504-70=\left(6x+2\right)^{2}$$

Use the distributive property to multiply $3x-1$ by $504$.

$$1512x-504-70=\left(6x+2\right)^{2}$$

Subtract $70$ from $-504$ to get $-574$.

$$1512x-574=\left(6x+2\right)^{2}$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(6x+2\right)^{2}$.

$$1512x-574=36x^{2}+24x+4$$

Subtract $36x^{2}$ from both sides.

$$1512x-574-36x^{2}=24x+4$$

Subtract $24x$ from both sides.

$$1512x-574-36x^{2}-24x=4$$

Combine $1512x$ and $-24x$ to get $1488x$.

$$1488x-574-36x^{2}=4$$

Add $574$ to both sides.

$$1488x-36x^{2}=4+574$$

Add $4$ and $574$ to get $578$.

$$1488x-36x^{2}=578$$

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 $x^{2}+bx=c$.

$$-36x^{2}+1488x=578$$

Divide both sides by $-36$.

$$\frac{-36x^{2}+1488x}{-36}=\frac{578}{-36}$$

Dividing by $-36$ undoes the multiplication by $-36$.

$$x^{2}+\frac{1488}{-36}x=\frac{578}{-36}$$

Reduce the fraction $\frac{1488}{-36}$ to lowest terms by extracting and canceling out $12$.

$$x^{2}-\frac{124}{3}x=\frac{578}{-36}$$

Reduce the fraction $\frac{578}{-36}$ to lowest terms by extracting and canceling out $2$.

$$x^{2}-\frac{124}{3}x=-\frac{289}{18}$$

Divide $-\frac{124}{3}$, the coefficient of the $x$ term, by $2$ to get $-\frac{62}{3}$. Then add the square of $-\frac{62}{3}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{124}{3}x+\left(-\frac{62}{3}\right)^{2}=-\frac{289}{18}+\left(-\frac{62}{3}\right)^{2}$$

Square $-\frac{62}{3}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{124}{3}x+\frac{3844}{9}=-\frac{289}{18}+\frac{3844}{9}$$

Add $-\frac{289}{18}$ to $\frac{3844}{9}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{124}{3}x+\frac{3844}{9}=\frac{7399}{18}$$

Factor $x^{2}-\frac{124}{3}x+\frac{3844}{9}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{62}{3}\right)^{2}=\frac{7399}{18}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{62}{3}\right)^{2}}=\sqrt{\frac{7399}{18}}$$

Simplify.

$$x-\frac{62}{3}=\frac{7\sqrt{302}}{6}$$ $$x-\frac{62}{3}=-\frac{7\sqrt{302}}{6}$$

Add $\frac{62}{3}$ to both sides of the equation.

$$x=\frac{7\sqrt{302}}{6}+\frac{62}{3}$$ $$x=-\frac{7\sqrt{302}}{6}+\frac{62}{3}$$