Solve 424=4x^2+60x+30x

$$424=4 { x }^{ 2 } +60x+30x$$

Solve for x

$$x = -\frac{53}{2} = -26\frac{1}{2} = -26.5$$

$$x=4$$

Steps Using Factoring By Grouping

Combine 60x and 30x to get 90x.

$$424=4x^{2}+90x$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+90x=424$$

Subtract 424 from both sides.

$$4x^{2}+90x-424=0$$

Divide both sides by 2.

$$2x^{2}+45x-212=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-212. To find a and b, set up a system to be solved.

$$a+b=45 ab=2\left(-212\right)=-424$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -424.

$$-1,424 -2,212 -4,106 -8,53$$

Calculate the sum for each pair.

$$-1+424=423 -2+212=210 -4+106=102 -8+53=45$$

The solution is the pair that gives sum 45.

$$a=-8 b=53$$

Rewrite 2x^{2}+45x-212 as \left(2x^{2}-8x\right)+\left(53x-212\right).

$$\left(2x^{2}-8x\right)+\left(53x-212\right)$$

Factor out 2x in the first and 53 in the second group.

$$2x\left(x-4\right)+53\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(2x+53\right)$$

To find equation solutions, solve x-4=0 and 2x+53=0.

$$x=4 x=-\frac{53}{2}$$

Steps Using the Quadratic Formula

Combine 60x and 30x to get 90x.

$$424=4x^{2}+90x$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+90x=424$$

Subtract 424 from both sides.

$$4x^{2}+90x-424=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 90 for b, and -424 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-90±\sqrt{90^{2}-4\times 4\left(-424\right)}}{2\times 4}$$

Square 90.

$$x=\frac{-90±\sqrt{8100-4\times 4\left(-424\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-90±\sqrt{8100-16\left(-424\right)}}{2\times 4}$$

Multiply -16 times -424.

$$x=\frac{-90±\sqrt{8100+6784}}{2\times 4}$$

Add 8100 to 6784.

$$x=\frac{-90±\sqrt{14884}}{2\times 4}$$

Take the square root of 14884.

$$x=\frac{-90±122}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-90±122}{8}$$

Now solve the equation x=\frac{-90±122}{8} when ± is plus. Add -90 to 122.

$$x=\frac{32}{8}$$

Divide 32 by 8.

$$x=4$$

Now solve the equation x=\frac{-90±122}{8} when ± is minus. Subtract 122 from -90.

$$x=-\frac{212}{8}$$

Reduce the fraction \frac{-212}{8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{53}{2}$$

The equation is now solved.

$$x=4 x=-\frac{53}{2}$$

Steps for Completing the Square

Combine 60x and 30x to get 90x.

$$424=4x^{2}+90x$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+90x=424$$

Divide both sides by 4.

$$\frac{4x^{2}+90x}{4}=\frac{424}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{90}{4}x=\frac{424}{4}$$

Reduce the fraction \frac{90}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{45}{2}x=\frac{424}{4}$$

Divide 424 by 4.

$$x^{2}+\frac{45}{2}x=106$$

Divide \frac{45}{2}, the coefficient of the x term, by 2 to get \frac{45}{4}. Then add the square of \frac{45}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{45}{2}x+\left(\frac{45}{4}\right)^{2}=106+\left(\frac{45}{4}\right)^{2}$$

Square \frac{45}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{45}{2}x+\frac{2025}{16}=106+\frac{2025}{16}$$

Add 106 to \frac{2025}{16}.

$$x^{2}+\frac{45}{2}x+\frac{2025}{16}=\frac{3721}{16}$$

Factor x^{2}+\frac{45}{2}x+\frac{2025}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{45}{4}\right)^{2}=\frac{3721}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{45}{4}\right)^{2}}=\sqrt{\frac{3721}{16}}$$

Simplify.

$$x+\frac{45}{4}=\frac{61}{4} x+\frac{45}{4}=-\frac{61}{4}$$

Subtract \frac{45}{4} from both sides of the equation.

$$x=4 x=-\frac{53}{2}$$