Solve Write the quadratic equation in standard form: - 4x - 19 = x ^ 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

$$\left. \begin{array} { l } { \quad - 4 x - 19 = x ^ { 2 } } \end{array} \right.$$

Solve for x

$x=-\sqrt{15}i-2\approx -2-3.872983346i$
$x=-2+\sqrt{15}i\approx -2+3.872983346i$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

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

$$-4x-19-x^{2}=0$$

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.

$$-x^{2}-4x-19=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-1$ for $a$, $-4$ for $b$, and $-19$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\left(-19\right)}}{2\left(-1\right)}$$

Square $-4$.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\left(-19\right)}}{2\left(-1\right)}$$

Multiply $-4$ times $-1$.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\left(-19\right)}}{2\left(-1\right)}$$

Multiply $4$ times $-19$.

$$x=\frac{-\left(-4\right)±\sqrt{16-76}}{2\left(-1\right)}$$

Add $16$ to $-76$.

$$x=\frac{-\left(-4\right)±\sqrt{-60}}{2\left(-1\right)}$$

Take the square root of $-60$.

$$x=\frac{-\left(-4\right)±2\sqrt{15}i}{2\left(-1\right)}$$

The opposite of $-4$ is $4$.

$$x=\frac{4±2\sqrt{15}i}{2\left(-1\right)}$$

Multiply $2$ times $-1$.

$$x=\frac{4±2\sqrt{15}i}{-2}$$

Now solve the equation $x=\frac{4±2\sqrt{15}i}{-2}$ when $±$ is plus. Add $4$ to $2i\sqrt{15}$.

$$x=\frac{4+2\sqrt{15}i}{-2}$$

Divide $4+2i\sqrt{15}$ by $-2$.

$$x=-\sqrt{15}i-2$$

Now solve the equation $x=\frac{4±2\sqrt{15}i}{-2}$ when $±$ is minus. Subtract $2i\sqrt{15}$ from $4$.

$$x=\frac{-2\sqrt{15}i+4}{-2}$$

Divide $4-2i\sqrt{15}$ by $-2$.

$$x=-2+\sqrt{15}i$$

The equation is now solved.

$$x=-\sqrt{15}i-2$$ $$x=-2+\sqrt{15}i$$

Steps for Completing the Square

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

$$-4x-19-x^{2}=0$$

Add $19$ to both sides. Anything plus zero gives itself.

$$-4x-x^{2}=19$$

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

$$-x^{2}-4x=19$$

Divide both sides by $-1$.

$$\frac{-x^{2}-4x}{-1}=\frac{19}{-1}$$

Dividing by $-1$ undoes the multiplication by $-1$.

$$x^{2}+\left(-\frac{4}{-1}\right)x=\frac{19}{-1}$$

Divide $-4$ by $-1$.

$$x^{2}+4x=\frac{19}{-1}$$

Divide $19$ by $-1$.

$$x^{2}+4x=-19$$

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

$$x^{2}+4x+2^{2}=-19+2^{2}$$

Square $2$.

$$x^{2}+4x+4=-19+4$$

Add $-19$ to $4$.

$$x^{2}+4x+4=-15$$

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

$$\left(x+2\right)^{2}=-15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{-15}$$

Simplify.

$$x+2=\sqrt{15}i$$ $$x+2=-\sqrt{15}i$$

Subtract $2$ from both sides of the equation.

$$x=-2+\sqrt{15}i$$ $$x=-\sqrt{15}i-2$$