Solve 4x^2+25x=-36

$$4 x ^ { 2 } + 25 x = - 36$$

Solve for x

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

$$x=-4$$

Steps Using Factoring By Grouping

Add 36 to both sides.

$$4x^{2}+25x+36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+36. To find a and b, set up a system to be solved.

$$a+b=25 ab=4\times 36=144$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Calculate the sum for each pair.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

The solution is the pair that gives sum 25.

$$a=9 b=16$$

Rewrite 4x^{2}+25x+36 as \left(4x^{2}+9x\right)+\left(16x+36\right).

$$\left(4x^{2}+9x\right)+\left(16x+36\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(4x+9\right)+4\left(4x+9\right)$$

Factor out common term 4x+9 by using distributive property.

$$\left(4x+9\right)\left(x+4\right)$$

To find equation solutions, solve 4x+9=0 and x+4=0.

$$x=-\frac{9}{4} x=-4$$

Steps Using the Quadratic Formula

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.

$$4x^{2}+25x=-36$$

Add 36 to both sides of the equation.

$$4x^{2}+25x-\left(-36\right)=-36-\left(-36\right)$$

Subtracting -36 from itself leaves 0.

$$4x^{2}+25x-\left(-36\right)=0$$

Subtract -36 from 0.

$$4x^{2}+25x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 25 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 4\times 36}}{2\times 4}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 4\times 36}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-25±\sqrt{625-16\times 36}}{2\times 4}$$

Multiply -16 times 36.

$$x=\frac{-25±\sqrt{625-576}}{2\times 4}$$

Add 625 to -576.

$$x=\frac{-25±\sqrt{49}}{2\times 4}$$

Take the square root of 49.

$$x=\frac{-25±7}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-25±7}{8}$$

Now solve the equation x=\frac{-25±7}{8} when ± is plus. Add -25 to 7.

$$x=-\frac{18}{8}$$

Reduce the fraction \frac{-18}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{9}{4}$$

Now solve the equation x=\frac{-25±7}{8} when ± is minus. Subtract 7 from -25.

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

Divide -32 by 8.

$$x=-4$$

The equation is now solved.

$$x=-\frac{9}{4} x=-4$$

Steps for Completing the Square

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.

$$4x^{2}+25x=-36$$

Divide both sides by 4.

$$\frac{4x^{2}+25x}{4}=-\frac{36}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{25}{4}x=-\frac{36}{4}$$

Divide -36 by 4.

$$x^{2}+\frac{25}{4}x=-9$$

Divide \frac{25}{4}, the coefficient of the x term, by 2 to get \frac{25}{8}. Then add the square of \frac{25}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{25}{4}x+\left(\frac{25}{8}\right)^{2}=-9+\left(\frac{25}{8}\right)^{2}$$

Square \frac{25}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{25}{4}x+\frac{625}{64}=-9+\frac{625}{64}$$

Add -9 to \frac{625}{64}.

$$x^{2}+\frac{25}{4}x+\frac{625}{64}=\frac{49}{64}$$

Factor x^{2}+\frac{25}{4}x+\frac{625}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{8}\right)^{2}=\frac{49}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Simplify.

$$x+\frac{25}{8}=\frac{7}{8} x+\frac{25}{8}=-\frac{7}{8}$$

Subtract \frac{25}{8} from both sides of the equation.

$$x=-\frac{9}{4} x=-4$$