Solve 16x^2-25x=-9

$$16 { x }^{ 2 } -25x=-9$$

Solve for x

$$x=1$$

$$x=\frac{9}{16}=0.5625$$

Steps Using Factoring By Grouping

Add 9 to both sides.

$$16x^{2}-25x+9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 16x^{2}+ax+bx+9. To find a and b, set up a system to be solved.

$$a+b=-25 ab=16\times 9=144$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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.

$$a=-16 b=-9$$

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

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

Factor out 16x in the first and -9 in the second group.

$$16x\left(x-1\right)-9\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(16x-9\right)$$

To find equation solutions, solve x-1=0 and 16x-9=0.

$$x=1 x=\frac{9}{16}$$

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.

$$16x^{2}-25x=-9$$

Add 9 to both sides of the equation.

$$16x^{2}-25x-\left(-9\right)=-9-\left(-9\right)$$

Subtracting -9 from itself leaves 0.

$$16x^{2}-25x-\left(-9\right)=0$$

Subtract -9 from 0.

$$16x^{2}-25x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -25 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 16\times 9}}{2\times 16}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 16\times 9}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-25\right)±\sqrt{625-64\times 9}}{2\times 16}$$

Multiply -64 times 9.

$$x=\frac{-\left(-25\right)±\sqrt{625-576}}{2\times 16}$$

Add 625 to -576.

$$x=\frac{-\left(-25\right)±\sqrt{49}}{2\times 16}$$

Take the square root of 49.

$$x=\frac{-\left(-25\right)±7}{2\times 16}$$

The opposite of -25 is 25.

$$x=\frac{25±7}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{25±7}{32}$$

Now solve the equation x=\frac{25±7}{32} when ± is plus. Add 25 to 7.

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

Divide 32 by 32.

$$x=1$$

Now solve the equation x=\frac{25±7}{32} when ± is minus. Subtract 7 from 25.

$$x=\frac{18}{32}$$

Reduce the fraction \frac{18}{32} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{16}$$

The equation is now solved.

$$x=1 x=\frac{9}{16}$$

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.

$$16x^{2}-25x=-9$$

Divide both sides by 16.

$$\frac{16x^{2}-25x}{16}=-\frac{9}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}-\frac{25}{16}x=-\frac{9}{16}$$

Divide -\frac{25}{16}, the coefficient of the x term, by 2 to get -\frac{25}{32}. Then add the square of -\frac{25}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{16}x+\left(-\frac{25}{32}\right)^{2}=-\frac{9}{16}+\left(-\frac{25}{32}\right)^{2}$$

Square -\frac{25}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{16}x+\frac{625}{1024}=-\frac{9}{16}+\frac{625}{1024}$$

Add -\frac{9}{16} to \frac{625}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{16}x+\frac{625}{1024}=\frac{49}{1024}$$

Factor x^{2}-\frac{25}{16}x+\frac{625}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{32}\right)^{2}=\frac{49}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{32}\right)^{2}}=\sqrt{\frac{49}{1024}}$$

Simplify.

$$x-\frac{25}{32}=\frac{7}{32} x-\frac{25}{32}=-\frac{7}{32}$$

Add \frac{25}{32} to both sides of the equation.

$$x=1 x=\frac{9}{16}$$