Solve 10(x)^(2)+23\div6-2(x+1)^(2)=19-8x\div2

Question

$$10 { x }^{ 2 } +23 \div 6-2 { \left(x+1 \right) }^{ 2 } =19-8x \div 2$$

Solve for x

$x = \frac{\sqrt{309}}{12} \approx 1.464866319$
$x = -\frac{\sqrt{309}}{12} \approx -1.464866319$

Steps by Finding Square Root

Steps Using the Quadratic Formula

Steps by Finding Square Root

Multiply both sides of the equation by $6$, the least common multiple of $6,2$.

$$6\left(10x^{2}+\frac{23}{6}\right)-12\left(x+1\right)^{2}=114-3\times 8x$$

Use the distributive property to multiply $6$ by $10x^{2}+\frac{23}{6}$.

$$60x^{2}+23-12\left(x+1\right)^{2}=114-3\times 8x$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(x+1\right)^{2}$.

$$60x^{2}+23-12\left(x^{2}+2x+1\right)=114-3\times 8x$$

Use the distributive property to multiply $-12$ by $x^{2}+2x+1$.

$$60x^{2}+23-12x^{2}-24x-12=114-3\times 8x$$

Combine $60x^{2}$ and $-12x^{2}$ to get $48x^{2}$.

$$48x^{2}+23-24x-12=114-3\times 8x$$

Subtract $12$ from $23$ to get $11$.

$$48x^{2}+11-24x=114-3\times 8x$$

Multiply $-3$ and $8$ to get $-24$.

$$48x^{2}+11-24x=114-24x$$

Add $24x$ to both sides.

$$48x^{2}+11-24x+24x=114$$

Combine $-24x$ and $24x$ to get $0$.

$$48x^{2}+11=114$$

Subtract $11$ from both sides.

$$48x^{2}=114-11$$

Subtract $11$ from $114$ to get $103$.

$$48x^{2}=103$$

Divide both sides by $48$.

$$x^{2}=\frac{103}{48}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{309}}{12}$$ $$x=-\frac{\sqrt{309}}{12}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by $6$, the least common multiple of $6,2$.

$$6\left(10x^{2}+\frac{23}{6}\right)-12\left(x+1\right)^{2}=114-3\times 8x$$

Use the distributive property to multiply $6$ by $10x^{2}+\frac{23}{6}$.

$$60x^{2}+23-12\left(x+1\right)^{2}=114-3\times 8x$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(x+1\right)^{2}$.

$$60x^{2}+23-12\left(x^{2}+2x+1\right)=114-3\times 8x$$

Use the distributive property to multiply $-12$ by $x^{2}+2x+1$.

$$60x^{2}+23-12x^{2}-24x-12=114-3\times 8x$$

Combine $60x^{2}$ and $-12x^{2}$ to get $48x^{2}$.

$$48x^{2}+23-24x-12=114-3\times 8x$$

Subtract $12$ from $23$ to get $11$.

$$48x^{2}+11-24x=114-3\times 8x$$

Multiply $-3$ and $8$ to get $-24$.

$$48x^{2}+11-24x=114-24x$$

Subtract $114$ from both sides.

$$48x^{2}+11-24x-114=-24x$$

Subtract $114$ from $11$ to get $-103$.

$$48x^{2}-103-24x=-24x$$

Add $24x$ to both sides.

$$48x^{2}-103-24x+24x=0$$

Combine $-24x$ and $24x$ to get $0$.

$$48x^{2}-103=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $48$ for $a$, $0$ for $b$, and $-103$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{0±\sqrt{0^{2}-4\times 48\left(-103\right)}}{2\times 48}$$

Square $0$.

$$x=\frac{0±\sqrt{-4\times 48\left(-103\right)}}{2\times 48}$$

Multiply $-4$ times $48$.

$$x=\frac{0±\sqrt{-192\left(-103\right)}}{2\times 48}$$

Multiply $-192$ times $-103$.

$$x=\frac{0±\sqrt{19776}}{2\times 48}$$

Take the square root of $19776$.

$$x=\frac{0±8\sqrt{309}}{2\times 48}$$

Multiply $2$ times $48$.

$$x=\frac{0±8\sqrt{309}}{96}$$

Now solve the equation $x=\frac{0±8\sqrt{309}}{96}$ when $±$ is plus.

$$x=\frac{\sqrt{309}}{12}$$

Now solve the equation $x=\frac{0±8\sqrt{309}}{96}$ when $±$ is minus.

$$x=-\frac{\sqrt{309}}{12}$$

The equation is now solved.

$$x=\frac{\sqrt{309}}{12}$$ $$x=-\frac{\sqrt{309}}{12}$$

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Example

Question

Solve for x

Steps by Finding Square Root

Steps Using the Quadratic Formula

Steps by Finding Square Root

Steps Using the Quadratic Formula