Solve left(5-xright)^2-2(5-x)+41+9x^2-34x-6(5-x)x=0

$${ \left(5-x \right) }^{ 2 } -2(5-x)+41+9 { x }^{ 2 } -34x-6(5-x)x=0$$

Solve for x

$$x=1$$

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$25-10x+x^{2}-2\left(5-x\right)+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Use the distributive property to multiply -2 by 5-x.

$$25-10x+x^{2}-10+2x+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Subtract 10 from 25 to get 15.

$$15-10x+x^{2}+2x+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Combine -10x and 2x to get -8x.

$$15-8x+x^{2}+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Add 15 and 41 to get 56.

$$56-8x+x^{2}+9x^{2}-34x-6\left(5-x\right)x=0$$

Combine x^{2} and 9x^{2} to get 10x^{2}.

$$56-8x+10x^{2}-34x-6\left(5-x\right)x=0$$

Combine -8x and -34x to get -42x.

$$56-42x+10x^{2}-6\left(5-x\right)x=0$$

Use the distributive property to multiply -6 by 5-x.

$$56-42x+10x^{2}+\left(-30+6x\right)x=0$$

Use the distributive property to multiply -30+6x by x.

$$56-42x+10x^{2}-30x+6x^{2}=0$$

Combine -42x and -30x to get -72x.

$$56-72x+10x^{2}+6x^{2}=0$$

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

$$56-72x+16x^{2}=0$$

Divide both sides by 8.

$$7-9x+2x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$2x^{2}-9x+7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=-9 ab=2\times 7=14$$

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 14.

$$-1,-14 -2,-7$$

Calculate the sum for each pair.

$$-1-14=-15 -2-7=-9$$

The solution is the pair that gives sum -9.

$$a=-7 b=-2$$

Rewrite 2x^{2}-9x+7 as \left(2x^{2}-7x\right)+\left(-2x+7\right).

$$\left(2x^{2}-7x\right)+\left(-2x+7\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(2x-7\right)-\left(2x-7\right)$$

Factor out common term 2x-7 by using distributive property.

$$\left(2x-7\right)\left(x-1\right)$$

To find equation solutions, solve 2x-7=0 and x-1=0.

$$x=\frac{7}{2} x=1$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$25-10x+x^{2}-2\left(5-x\right)+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Use the distributive property to multiply -2 by 5-x.

$$25-10x+x^{2}-10+2x+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Subtract 10 from 25 to get 15.

$$15-10x+x^{2}+2x+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Combine -10x and 2x to get -8x.

$$15-8x+x^{2}+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Add 15 and 41 to get 56.

$$56-8x+x^{2}+9x^{2}-34x-6\left(5-x\right)x=0$$

Combine x^{2} and 9x^{2} to get 10x^{2}.

$$56-8x+10x^{2}-34x-6\left(5-x\right)x=0$$

Combine -8x and -34x to get -42x.

$$56-42x+10x^{2}-6\left(5-x\right)x=0$$

Use the distributive property to multiply -6 by 5-x.

$$56-42x+10x^{2}+\left(-30+6x\right)x=0$$

Use the distributive property to multiply -30+6x by x.

$$56-42x+10x^{2}-30x+6x^{2}=0$$

Combine -42x and -30x to get -72x.

$$56-72x+10x^{2}+6x^{2}=0$$

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

$$56-72x+16x^{2}=0$$

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.

$$16x^{2}-72x+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -72 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 16\times 56}}{2\times 16}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 16\times 56}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-72\right)±\sqrt{5184-64\times 56}}{2\times 16}$$

Multiply -64 times 56.

$$x=\frac{-\left(-72\right)±\sqrt{5184-3584}}{2\times 16}$$

Add 5184 to -3584.

$$x=\frac{-\left(-72\right)±\sqrt{1600}}{2\times 16}$$

Take the square root of 1600.

$$x=\frac{-\left(-72\right)±40}{2\times 16}$$

The opposite of -72 is 72.

$$x=\frac{72±40}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{72±40}{32}$$

Now solve the equation x=\frac{72±40}{32} when ± is plus. Add 72 to 40.

$$x=\frac{112}{32}$$

Reduce the fraction \frac{112}{32} to lowest terms by extracting and canceling out 16.

$$x=\frac{7}{2}$$

Now solve the equation x=\frac{72±40}{32} when ± is minus. Subtract 40 from 72.

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

Divide 32 by 32.

$$x=1$$

The equation is now solved.

$$x=\frac{7}{2} x=1$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$25-10x+x^{2}-2\left(5-x\right)+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Use the distributive property to multiply -2 by 5-x.

$$25-10x+x^{2}-10+2x+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Subtract 10 from 25 to get 15.

$$15-10x+x^{2}+2x+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Combine -10x and 2x to get -8x.

$$15-8x+x^{2}+41+9x^{2}-34x-6\left(5-x\right)x=0$$

Add 15 and 41 to get 56.

$$56-8x+x^{2}+9x^{2}-34x-6\left(5-x\right)x=0$$

Combine x^{2} and 9x^{2} to get 10x^{2}.

$$56-8x+10x^{2}-34x-6\left(5-x\right)x=0$$

Combine -8x and -34x to get -42x.

$$56-42x+10x^{2}-6\left(5-x\right)x=0$$

Use the distributive property to multiply -6 by 5-x.

$$56-42x+10x^{2}+\left(-30+6x\right)x=0$$

Use the distributive property to multiply -30+6x by x.

$$56-42x+10x^{2}-30x+6x^{2}=0$$

Combine -42x and -30x to get -72x.

$$56-72x+10x^{2}+6x^{2}=0$$

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

$$56-72x+16x^{2}=0$$

Subtract 56 from both sides. Anything subtracted from zero gives its negation.

$$-72x+16x^{2}=-56$$

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.

$$16x^{2}-72x=-56$$

Divide both sides by 16.

$$\frac{16x^{2}-72x}{16}=-\frac{56}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{72}{16}\right)x=-\frac{56}{16}$$

Reduce the fraction \frac{-72}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{9}{2}x=-\frac{56}{16}$$

Reduce the fraction \frac{-56}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{9}{2}x=-\frac{7}{2}$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-\frac{7}{2}+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-\frac{7}{2}+\frac{81}{16}$$

Add -\frac{7}{2} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{25}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{9}{4}=\frac{5}{4} x-\frac{9}{4}=-\frac{5}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=\frac{7}{2} x=1$$