Solve 10x^2-11x+9=13x-6x^2

$$10 { x }^{ 2 } -11x+9=13x-6 { x }^{ 2 }$$

Solve for x

$$x=\frac{3}{4}=0.75$$

Steps Using Factoring By Grouping

Subtract 13x from both sides.

$$10x^{2}-11x+9-13x=-6x^{2}$$

Combine -11x and -13x to get -24x.

$$10x^{2}-24x+9=-6x^{2}$$

Add 6x^{2} to both sides.

$$10x^{2}-24x+9+6x^{2}=0$$

Combine 10x^{2} and 6x^{2} to get 16x^{2}.

$$16x^{2}-24x+9=0$$

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.

$$a+b=-24 ab=16\times 9=144$$

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 -24.

$$a=-12 b=-12$$

Rewrite 16x^{2}-24x+9 as \left(16x^{2}-12x\right)+\left(-12x+9\right).

$$\left(16x^{2}-12x\right)+\left(-12x+9\right)$$

Factor out 4x in the first and -3 in the second group.

$$4x\left(4x-3\right)-3\left(4x-3\right)$$

Factor out common term 4x-3 by using distributive property.

$$\left(4x-3\right)\left(4x-3\right)$$

Rewrite as a binomial square.

$$\left(4x-3\right)^{2}$$

To find equation solution, solve 4x-3=0.

$$x=\frac{3}{4}$$

Steps Using the Quadratic Formula

Subtract 13x from both sides.

$$10x^{2}-11x+9-13x=-6x^{2}$$

Combine -11x and -13x to get -24x.

$$10x^{2}-24x+9=-6x^{2}$$

Add 6x^{2} to both sides.

$$10x^{2}-24x+9+6x^{2}=0$$

Combine 10x^{2} and 6x^{2} to get 16x^{2}.

$$16x^{2}-24x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -24 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 16\times 9}}{2\times 16}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 16\times 9}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-24\right)±\sqrt{576-64\times 9}}{2\times 16}$$

Multiply -64 times 9.

$$x=\frac{-\left(-24\right)±\sqrt{576-576}}{2\times 16}$$

Add 576 to -576.

$$x=\frac{-\left(-24\right)±\sqrt{0}}{2\times 16}$$

Take the square root of 0.

$$x=-\frac{-24}{2\times 16}$$

The opposite of -24 is 24.

$$x=\frac{24}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{24}{32}$$

Reduce the fraction \frac{24}{32} to lowest terms by extracting and canceling out 8.

$$x=\frac{3}{4}$$

Steps for Completing the Square

Subtract 13x from both sides.

$$10x^{2}-11x+9-13x=-6x^{2}$$

Combine -11x and -13x to get -24x.

$$10x^{2}-24x+9=-6x^{2}$$

Add 6x^{2} to both sides.

$$10x^{2}-24x+9+6x^{2}=0$$

Combine 10x^{2} and 6x^{2} to get 16x^{2}.

$$16x^{2}-24x+9=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$16x^{2}-24x=-9$$

Divide both sides by 16.

$$\frac{16x^{2}-24x}{16}=-\frac{9}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{24}{16}\right)x=-\frac{9}{16}$$

Reduce the fraction \frac{-24}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{3}{2}x=-\frac{9}{16}$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{9}{16}+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{-9+9}{16}$$

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

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=0$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-\frac{3}{4}=0 x-\frac{3}{4}=0$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{3}{4} x=\frac{3}{4}$$

The equation is now solved. Solutions are the same.

$$x=\frac{3}{4}$$