Solve 2 * 22/7 * x(x + 21) = 1232 | Equatix - Math Solver

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Example

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Question

$$2\times\frac{22}{7}\times x(x+21)=1232$$

Solve for x

$x=-28$
$x=7$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Express $2\times \frac{22}{7}$ as a single fraction.

$$\frac{2\times 22}{7}x\left(x+21\right)=1232$$

Multiply $2$ and $22$ to get $44$.

$$\frac{44}{7}x\left(x+21\right)=1232$$

Use the distributive property to multiply $\frac{44}{7}x$ by $x+21$.

$$\frac{44}{7}xx+\frac{44}{7}x\times 21=1232$$

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

$$\frac{44}{7}x^{2}+\frac{44}{7}x\times 21=1232$$

Express $\frac{44}{7}\times 21$ as a single fraction.

$$\frac{44}{7}x^{2}+\frac{44\times 21}{7}x=1232$$

Multiply $44$ and $21$ to get $924$.

$$\frac{44}{7}x^{2}+\frac{924}{7}x=1232$$

Divide $924$ by $7$ to get $132$.

$$\frac{44}{7}x^{2}+132x=1232$$

Subtract $1232$ from both sides.

$$\frac{44}{7}x^{2}+132x-1232=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $\frac{44}{7}$ for $a$, $132$ for $b$, and $-1232$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-132±\sqrt{132^{2}-4\times \frac{44}{7}\left(-1232\right)}}{2\times \frac{44}{7}}$$

Square $132$.

$$x=\frac{-132±\sqrt{17424-4\times \frac{44}{7}\left(-1232\right)}}{2\times \frac{44}{7}}$$

Multiply $-4$ times $\frac{44}{7}$.

$$x=\frac{-132±\sqrt{17424-\frac{176}{7}\left(-1232\right)}}{2\times \frac{44}{7}}$$

Multiply $-\frac{176}{7}$ times $-1232$.

$$x=\frac{-132±\sqrt{17424+30976}}{2\times \frac{44}{7}}$$

Add $17424$ to $30976$.

$$x=\frac{-132±\sqrt{48400}}{2\times \frac{44}{7}}$$

Take the square root of $48400$.

$$x=\frac{-132±220}{2\times \frac{44}{7}}$$

Multiply $2$ times $\frac{44}{7}$.

$$x=\frac{-132±220}{\frac{88}{7}}$$

Now solve the equation $x=\frac{-132±220}{\frac{88}{7}}$ when $±$ is plus. Add $-132$ to $220$.

$$x=\frac{88}{\frac{88}{7}}$$

Divide $88$ by $\frac{88}{7}$ by multiplying $88$ by the reciprocal of $\frac{88}{7}$.

$$x=7$$

Now solve the equation $x=\frac{-132±220}{\frac{88}{7}}$ when $±$ is minus. Subtract $220$ from $-132$.

$$x=-\frac{352}{\frac{88}{7}}$$

Divide $-352$ by $\frac{88}{7}$ by multiplying $-352$ by the reciprocal of $\frac{88}{7}$.

$$x=-28$$

The equation is now solved.

$$x=7$$ $$x=-28$$

Steps for Completing the Square

Express $2\times \frac{22}{7}$ as a single fraction.

$$\frac{2\times 22}{7}x\left(x+21\right)=1232$$

Multiply $2$ and $22$ to get $44$.

$$\frac{44}{7}x\left(x+21\right)=1232$$

Use the distributive property to multiply $\frac{44}{7}x$ by $x+21$.

$$\frac{44}{7}xx+\frac{44}{7}x\times 21=1232$$

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

$$\frac{44}{7}x^{2}+\frac{44}{7}x\times 21=1232$$

Express $\frac{44}{7}\times 21$ as a single fraction.

$$\frac{44}{7}x^{2}+\frac{44\times 21}{7}x=1232$$

Multiply $44$ and $21$ to get $924$.

$$\frac{44}{7}x^{2}+\frac{924}{7}x=1232$$

Divide $924$ by $7$ to get $132$.

$$\frac{44}{7}x^{2}+132x=1232$$

Divide both sides of the equation by $\frac{44}{7}$, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{44}{7}x^{2}+132x}{\frac{44}{7}}=\frac{1232}{\frac{44}{7}}$$

Dividing by $\frac{44}{7}$ undoes the multiplication by $\frac{44}{7}$.

$$x^{2}+\frac{132}{\frac{44}{7}}x=\frac{1232}{\frac{44}{7}}$$

Divide $132$ by $\frac{44}{7}$ by multiplying $132$ by the reciprocal of $\frac{44}{7}$.

$$x^{2}+21x=\frac{1232}{\frac{44}{7}}$$

Divide $1232$ by $\frac{44}{7}$ by multiplying $1232$ by the reciprocal of $\frac{44}{7}$.

$$x^{2}+21x=196$$

Divide $21$, the coefficient of the $x$ term, by $2$ to get $\frac{21}{2}$. Then add the square of $\frac{21}{2}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+21x+\left(\frac{21}{2}\right)^{2}=196+\left(\frac{21}{2}\right)^{2}$$

Square $\frac{21}{2}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}+21x+\frac{441}{4}=196+\frac{441}{4}$$

Add $196$ to $\frac{441}{4}$.

$$x^{2}+21x+\frac{441}{4}=\frac{1225}{4}$$

Factor $x^{2}+21x+\frac{441}{4}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x+\frac{21}{2}\right)^{2}=\frac{1225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{21}{2}\right)^{2}}=\sqrt{\frac{1225}{4}}$$

Simplify.

$$x+\frac{21}{2}=\frac{35}{2}$$ $$x+\frac{21}{2}=-\frac{35}{2}$$

Subtract $\frac{21}{2}$ from both sides of the equation.

$$x=7$$ $$x=-28$$