Solve 26x(2x-6/3=32x+1x^2-6

$$\frac { 26 x ( 2 x - 6 } { 3 } = 32 x + 1 x ^ { 2 } - 6$$

Solve for x

$$x = \frac{3 \sqrt{34} + 18}{7} \approx 5.070407955$$

$$x=\frac{18-3\sqrt{34}}{7}\approx 0.072449188$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$26x\left(2x-6\right)=96x+3x^{2}-18$$

Use the distributive property to multiply 26x by 2x-6.

$$52x^{2}-156x=96x+3x^{2}-18$$

Subtract 96x from both sides.

$$52x^{2}-156x-96x=3x^{2}-18$$

Combine -156x and -96x to get -252x.

$$52x^{2}-252x=3x^{2}-18$$

Subtract 3x^{2} from both sides.

$$52x^{2}-252x-3x^{2}=-18$$

Combine 52x^{2} and -3x^{2} to get 49x^{2}.

$$49x^{2}-252x=-18$$

Add 18 to both sides.

$$49x^{2}-252x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 49 for a, -252 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-252\right)±\sqrt{\left(-252\right)^{2}-4\times 49\times 18}}{2\times 49}$$

Square -252.

$$x=\frac{-\left(-252\right)±\sqrt{63504-4\times 49\times 18}}{2\times 49}$$

Multiply -4 times 49.

$$x=\frac{-\left(-252\right)±\sqrt{63504-196\times 18}}{2\times 49}$$

Multiply -196 times 18.

$$x=\frac{-\left(-252\right)±\sqrt{63504-3528}}{2\times 49}$$

Add 63504 to -3528.

$$x=\frac{-\left(-252\right)±\sqrt{59976}}{2\times 49}$$

Take the square root of 59976.

$$x=\frac{-\left(-252\right)±42\sqrt{34}}{2\times 49}$$

The opposite of -252 is 252.

$$x=\frac{252±42\sqrt{34}}{2\times 49}$$

Multiply 2 times 49.

$$x=\frac{252±42\sqrt{34}}{98}$$

Now solve the equation x=\frac{252±42\sqrt{34}}{98} when ± is plus. Add 252 to 42\sqrt{34}.

$$x=\frac{42\sqrt{34}+252}{98}$$

Divide 252+42\sqrt{34} by 98.

$$x=\frac{3\sqrt{34}+18}{7}$$

Now solve the equation x=\frac{252±42\sqrt{34}}{98} when ± is minus. Subtract 42\sqrt{34} from 252.

$$x=\frac{252-42\sqrt{34}}{98}$$

Divide 252-42\sqrt{34} by 98.

$$x=\frac{18-3\sqrt{34}}{7}$$

The equation is now solved.

$$x=\frac{3\sqrt{34}+18}{7} x=\frac{18-3\sqrt{34}}{7}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$26x\left(2x-6\right)=96x+3x^{2}-18$$

Use the distributive property to multiply 26x by 2x-6.

$$52x^{2}-156x=96x+3x^{2}-18$$

Subtract 96x from both sides.

$$52x^{2}-156x-96x=3x^{2}-18$$

Combine -156x and -96x to get -252x.

$$52x^{2}-252x=3x^{2}-18$$

Subtract 3x^{2} from both sides.

$$52x^{2}-252x-3x^{2}=-18$$

Combine 52x^{2} and -3x^{2} to get 49x^{2}.

$$49x^{2}-252x=-18$$

Divide both sides by 49.

$$\frac{49x^{2}-252x}{49}=-\frac{18}{49}$$

Dividing by 49 undoes the multiplication by 49.

$$x^{2}+\left(-\frac{252}{49}\right)x=-\frac{18}{49}$$

Reduce the fraction \frac{-252}{49} to lowest terms by extracting and canceling out 7.

$$x^{2}-\frac{36}{7}x=-\frac{18}{49}$$

Divide -\frac{36}{7}, the coefficient of the x term, by 2 to get -\frac{18}{7}. Then add the square of -\frac{18}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{36}{7}x+\left(-\frac{18}{7}\right)^{2}=-\frac{18}{49}+\left(-\frac{18}{7}\right)^{2}$$

Square -\frac{18}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{36}{7}x+\frac{324}{49}=\frac{-18+324}{49}$$

Add -\frac{18}{49} to \frac{324}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{36}{7}x+\frac{324}{49}=\frac{306}{49}$$

Factor x^{2}-\frac{36}{7}x+\frac{324}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{18}{7}\right)^{2}=\frac{306}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{18}{7}\right)^{2}}=\sqrt{\frac{306}{49}}$$

Simplify.

$$x-\frac{18}{7}=\frac{3\sqrt{34}}{7} x-\frac{18}{7}=-\frac{3\sqrt{34}}{7}$$

Add \frac{18}{7} to both sides of the equation.

$$x=\frac{3\sqrt{34}+18}{7} x=\frac{18-3\sqrt{34}}{7}$$