Solve (x-4)(x-4)-(4x+5)(3x-10)=17x-110quad.5?

$$( x - 4 ) ( x - 4 ) - ( 4 x + 5 ) ( 3 x - 10 ) = 17 x - 110 \quad .5 ?$$

Solve for x

$$x = \frac{\sqrt{7766}}{22} \approx 4.005677789$$

$$x = -\frac{\sqrt{7766}}{22} \approx -4.005677789$$

Steps by Finding Square Root

Multiply x-4 and x-4 to get \left(x-4\right)^{2}.

$$\left(x-4\right)^{2}-\left(4x+5\right)\left(3x-10\right)=17x-110.5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16-\left(4x+5\right)\left(3x-10\right)=17x-110.5$$

Use the distributive property to multiply 4x+5 by 3x-10 and combine like terms.

$$x^{2}-8x+16-\left(12x^{2}-25x-50\right)=17x-110.5$$

To find the opposite of 12x^{2}-25x-50, find the opposite of each term.

$$x^{2}-8x+16-12x^{2}+25x+50=17x-110.5$$

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$-11x^{2}-8x+16+25x+50=17x-110.5$$

Combine -8x and 25x to get 17x.

$$-11x^{2}+17x+16+50=17x-110.5$$

Add 16 and 50 to get 66.

$$-11x^{2}+17x+66=17x-110.5$$

Subtract 17x from both sides.

$$-11x^{2}+17x+66-17x=-110.5$$

Combine 17x and -17x to get 0.

$$-11x^{2}+66=-110.5$$

Subtract 66 from both sides.

$$-11x^{2}=-110.5-66$$

Subtract 66 from -110.5 to get -176.5.

$$-11x^{2}=-176.5$$

Divide both sides by -11.

$$x^{2}=\frac{-176.5}{-11}$$

Expand \frac{-176.5}{-11} by multiplying both numerator and the denominator by 10.

$$x^{2}=\frac{-1765}{-110}$$

Reduce the fraction \frac{-1765}{-110} to lowest terms by extracting and canceling out -5.

$$x^{2}=\frac{353}{22}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{7766}}{22} x=-\frac{\sqrt{7766}}{22}$$

Steps Using the Quadratic Formula

Multiply x-4 and x-4 to get \left(x-4\right)^{2}.

$$\left(x-4\right)^{2}-\left(4x+5\right)\left(3x-10\right)=17x-110.5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16-\left(4x+5\right)\left(3x-10\right)=17x-110.5$$

Use the distributive property to multiply 4x+5 by 3x-10 and combine like terms.

$$x^{2}-8x+16-\left(12x^{2}-25x-50\right)=17x-110.5$$

To find the opposite of 12x^{2}-25x-50, find the opposite of each term.

$$x^{2}-8x+16-12x^{2}+25x+50=17x-110.5$$

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$-11x^{2}-8x+16+25x+50=17x-110.5$$

Combine -8x and 25x to get 17x.

$$-11x^{2}+17x+16+50=17x-110.5$$

Add 16 and 50 to get 66.

$$-11x^{2}+17x+66=17x-110.5$$

Subtract 17x from both sides.

$$-11x^{2}+17x+66-17x=-110.5$$

Combine 17x and -17x to get 0.

$$-11x^{2}+66=-110.5$$

Add 110.5 to both sides.

$$-11x^{2}+66+110.5=0$$

Add 66 and 110.5 to get 176.5.

$$-11x^{2}+176.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -11 for a, 0 for b, and 176.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-11\right)\times 176.5}}{2\left(-11\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-11\right)\times 176.5}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$x=\frac{0±\sqrt{44\times 176.5}}{2\left(-11\right)}$$

Multiply 44 times 176.5.

$$x=\frac{0±\sqrt{7766}}{2\left(-11\right)}$$

Multiply 2 times -11.

$$x=\frac{0±\sqrt{7766}}{-22}$$

Now solve the equation x=\frac{0±\sqrt{7766}}{-22} when ± is plus.

$$x=-\frac{\sqrt{7766}}{22}$$

Now solve the equation x=\frac{0±\sqrt{7766}}{-22} when ± is minus.

$$x=\frac{\sqrt{7766}}{22}$$

The equation is now solved.

$$x=-\frac{\sqrt{7766}}{22} x=\frac{\sqrt{7766}}{22}$$