Solve left(8-3xright)^2=11-(3-2x)(4+x)

$${ \left(8-3x \right) }^{ 2 } =11-(3-2x)(4+x)$$

Solve for x

$$x = \frac{\sqrt{989} + 53}{14} \approx 6.032026456$$

$$x = \frac{53 - \sqrt{989}}{14} \approx 1.539402115$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-3x\right)^{2}.

$$64-48x+9x^{2}=11-\left(3-2x\right)\left(4+x\right)$$

Use the distributive property to multiply 3-2x by 4+x and combine like terms.

$$64-48x+9x^{2}=11-\left(12-5x-2x^{2}\right)$$

To find the opposite of 12-5x-2x^{2}, find the opposite of each term.

$$64-48x+9x^{2}=11-12+5x+2x^{2}$$

Subtract 12 from 11 to get -1.

$$64-48x+9x^{2}=-1+5x+2x^{2}$$

Subtract -1 from both sides.

$$64-48x+9x^{2}-\left(-1\right)=5x+2x^{2}$$

The opposite of -1 is 1.

$$64-48x+9x^{2}+1=5x+2x^{2}$$

Subtract 5x from both sides.

$$64-48x+9x^{2}+1-5x=2x^{2}$$

Add 64 and 1 to get 65.

$$65-48x+9x^{2}-5x=2x^{2}$$

Combine -48x and -5x to get -53x.

$$65-53x+9x^{2}=2x^{2}$$

Subtract 2x^{2} from both sides.

$$65-53x+9x^{2}-2x^{2}=0$$

Combine 9x^{2} and -2x^{2} to get 7x^{2}.

$$65-53x+7x^{2}=0$$

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.

$$7x^{2}-53x+65=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -53 for b, and 65 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-53\right)±\sqrt{\left(-53\right)^{2}-4\times 7\times 65}}{2\times 7}$$

Square -53.

$$x=\frac{-\left(-53\right)±\sqrt{2809-4\times 7\times 65}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-53\right)±\sqrt{2809-28\times 65}}{2\times 7}$$

Multiply -28 times 65.

$$x=\frac{-\left(-53\right)±\sqrt{2809-1820}}{2\times 7}$$

Add 2809 to -1820.

$$x=\frac{-\left(-53\right)±\sqrt{989}}{2\times 7}$$

The opposite of -53 is 53.

$$x=\frac{53±\sqrt{989}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{53±\sqrt{989}}{14}$$

Now solve the equation x=\frac{53±\sqrt{989}}{14} when ± is plus. Add 53 to \sqrt{989}.

$$x=\frac{\sqrt{989}+53}{14}$$

Now solve the equation x=\frac{53±\sqrt{989}}{14} when ± is minus. Subtract \sqrt{989} from 53.

$$x=\frac{53-\sqrt{989}}{14}$$

The equation is now solved.

$$x=\frac{\sqrt{989}+53}{14} x=\frac{53-\sqrt{989}}{14}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-3x\right)^{2}.

$$64-48x+9x^{2}=11-\left(3-2x\right)\left(4+x\right)$$

Use the distributive property to multiply 3-2x by 4+x and combine like terms.

$$64-48x+9x^{2}=11-\left(12-5x-2x^{2}\right)$$

To find the opposite of 12-5x-2x^{2}, find the opposite of each term.

$$64-48x+9x^{2}=11-12+5x+2x^{2}$$

Subtract 12 from 11 to get -1.

$$64-48x+9x^{2}=-1+5x+2x^{2}$$

Subtract 5x from both sides.

$$64-48x+9x^{2}-5x=-1+2x^{2}$$

Combine -48x and -5x to get -53x.

$$64-53x+9x^{2}=-1+2x^{2}$$

Subtract 2x^{2} from both sides.

$$64-53x+9x^{2}-2x^{2}=-1$$

Combine 9x^{2} and -2x^{2} to get 7x^{2}.

$$64-53x+7x^{2}=-1$$

Subtract 64 from both sides.

$$-53x+7x^{2}=-1-64$$

Subtract 64 from -1 to get -65.

$$-53x+7x^{2}=-65$$

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.

$$7x^{2}-53x=-65$$

Divide both sides by 7.

$$\frac{7x^{2}-53x}{7}=-\frac{65}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{53}{7}x=-\frac{65}{7}$$

Divide -\frac{53}{7}, the coefficient of the x term, by 2 to get -\frac{53}{14}. Then add the square of -\frac{53}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{53}{7}x+\left(-\frac{53}{14}\right)^{2}=-\frac{65}{7}+\left(-\frac{53}{14}\right)^{2}$$

Square -\frac{53}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{53}{7}x+\frac{2809}{196}=-\frac{65}{7}+\frac{2809}{196}$$

Add -\frac{65}{7} to \frac{2809}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{53}{7}x+\frac{2809}{196}=\frac{989}{196}$$

Factor x^{2}-\frac{53}{7}x+\frac{2809}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{53}{14}\right)^{2}=\frac{989}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{53}{14}\right)^{2}}=\sqrt{\frac{989}{196}}$$

Simplify.

$$x-\frac{53}{14}=\frac{\sqrt{989}}{14} x-\frac{53}{14}=-\frac{\sqrt{989}}{14}$$

Add \frac{53}{14} to both sides of the equation.

$$x=\frac{\sqrt{989}+53}{14} x=\frac{53-\sqrt{989}}{14}$$