Solve 25-8x+x^2=left(20x-35right)^2/49

$$25-8x+ { x }^{ 2 } = \frac{ { \left(20x-35 \right) }^{ 2 } }{ 49 }$$

Solve for x

$$x = \frac{112}{39} = 2\frac{34}{39} \approx 2.871794872$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 49.

$$1225-392x+49x^{2}=\left(20x-35\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(20x-35\right)^{2}.

$$1225-392x+49x^{2}=400x^{2}-1400x+1225$$

Subtract 400x^{2} from both sides.

$$1225-392x+49x^{2}-400x^{2}=-1400x+1225$$

Combine 49x^{2} and -400x^{2} to get -351x^{2}.

$$1225-392x-351x^{2}=-1400x+1225$$

Add 1400x to both sides.

$$1225-392x-351x^{2}+1400x=1225$$

Combine -392x and 1400x to get 1008x.

$$1225+1008x-351x^{2}=1225$$

Subtract 1225 from both sides.

$$1225+1008x-351x^{2}-1225=0$$

Subtract 1225 from 1225 to get 0.

$$1008x-351x^{2}=0$$

Factor out x.

$$x\left(1008-351x\right)=0$$

To find equation solutions, solve x=0 and 1008-351x=0.

$$x=0 x=\frac{112}{39}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 49.

$$1225-392x+49x^{2}=\left(20x-35\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(20x-35\right)^{2}.

$$1225-392x+49x^{2}=400x^{2}-1400x+1225$$

Subtract 400x^{2} from both sides.

$$1225-392x+49x^{2}-400x^{2}=-1400x+1225$$

Combine 49x^{2} and -400x^{2} to get -351x^{2}.

$$1225-392x-351x^{2}=-1400x+1225$$

Add 1400x to both sides.

$$1225-392x-351x^{2}+1400x=1225$$

Combine -392x and 1400x to get 1008x.

$$1225+1008x-351x^{2}=1225$$

Subtract 1225 from both sides.

$$1225+1008x-351x^{2}-1225=0$$

Subtract 1225 from 1225 to get 0.

$$1008x-351x^{2}=0$$

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.

$$-351x^{2}+1008x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -351 for a, 1008 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1008±\sqrt{1008^{2}}}{2\left(-351\right)}$$

Take the square root of 1008^{2}.

$$x=\frac{-1008±1008}{2\left(-351\right)}$$

Multiply 2 times -351.

$$x=\frac{-1008±1008}{-702}$$

Now solve the equation x=\frac{-1008±1008}{-702} when ± is plus. Add -1008 to 1008.

$$x=\frac{0}{-702}$$

Divide 0 by -702.

$$x=0$$

Now solve the equation x=\frac{-1008±1008}{-702} when ± is minus. Subtract 1008 from -1008.

$$x=-\frac{2016}{-702}$$

Reduce the fraction \frac{-2016}{-702} to lowest terms by extracting and canceling out 18.

$$x=\frac{112}{39}$$

The equation is now solved.

$$x=0 x=\frac{112}{39}$$

Steps for Completing the Square

Multiply both sides of the equation by 49.

$$1225-392x+49x^{2}=\left(20x-35\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(20x-35\right)^{2}.

$$1225-392x+49x^{2}=400x^{2}-1400x+1225$$

Subtract 400x^{2} from both sides.

$$1225-392x+49x^{2}-400x^{2}=-1400x+1225$$

Combine 49x^{2} and -400x^{2} to get -351x^{2}.

$$1225-392x-351x^{2}=-1400x+1225$$

Add 1400x to both sides.

$$1225-392x-351x^{2}+1400x=1225$$

Combine -392x and 1400x to get 1008x.

$$1225+1008x-351x^{2}=1225$$

Subtract 1225 from both sides.

$$1008x-351x^{2}=1225-1225$$

Subtract 1225 from 1225 to get 0.

$$1008x-351x^{2}=0$$

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.

$$-351x^{2}+1008x=0$$

Divide both sides by -351.

$$\frac{-351x^{2}+1008x}{-351}=\frac{0}{-351}$$

Dividing by -351 undoes the multiplication by -351.

$$x^{2}+\frac{1008}{-351}x=\frac{0}{-351}$$

Reduce the fraction \frac{1008}{-351} to lowest terms by extracting and canceling out 9.

$$x^{2}-\frac{112}{39}x=\frac{0}{-351}$$

Divide 0 by -351.

$$x^{2}-\frac{112}{39}x=0$$

Divide -\frac{112}{39}, the coefficient of the x term, by 2 to get -\frac{56}{39}. Then add the square of -\frac{56}{39} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{112}{39}x+\left(-\frac{56}{39}\right)^{2}=\left(-\frac{56}{39}\right)^{2}$$

Square -\frac{56}{39} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{112}{39}x+\frac{3136}{1521}=\frac{3136}{1521}$$

Factor x^{2}-\frac{112}{39}x+\frac{3136}{1521}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{56}{39}\right)^{2}=\frac{3136}{1521}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{56}{39}\right)^{2}}=\sqrt{\frac{3136}{1521}}$$

Simplify.

$$x-\frac{56}{39}=\frac{56}{39} x-\frac{56}{39}=-\frac{56}{39}$$

Add \frac{56}{39} to both sides of the equation.

$$x=\frac{112}{39} x=0$$