Solve 100x^2-176x-752=0

$$100 { x }^{ 2 } -176x-752=0$$

Solve for x

$$x=-2$$

$$x = \frac{94}{25} = 3\frac{19}{25} = 3.76$$

Steps Using Factoring By Grouping

Divide both sides by 4.

$$25x^{2}-44x-188=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 25x^{2}+ax+bx-188. To find a and b, set up a system to be solved.

$$a+b=-44 ab=25\left(-188\right)=-4700$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4700.

$$1,-4700 2,-2350 4,-1175 5,-940 10,-470 20,-235 25,-188 47,-100 50,-94$$

Calculate the sum for each pair.

$$1-4700=-4699 2-2350=-2348 4-1175=-1171 5-940=-935 10-470=-460 20-235=-215 25-188=-163 47-100=-53 50-94=-44$$

The solution is the pair that gives sum -44.

$$a=-94 b=50$$

Rewrite 25x^{2}-44x-188 as \left(25x^{2}-94x\right)+\left(50x-188\right).

$$\left(25x^{2}-94x\right)+\left(50x-188\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(25x-94\right)+2\left(25x-94\right)$$

Factor out common term 25x-94 by using distributive property.

$$\left(25x-94\right)\left(x+2\right)$$

To find equation solutions, solve 25x-94=0 and x+2=0.

$$x=\frac{94}{25} x=-2$$

Steps Using the Quadratic Formula

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.

$$100x^{2}-176x-752=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, -176 for b, and -752 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-176\right)±\sqrt{\left(-176\right)^{2}-4\times 100\left(-752\right)}}{2\times 100}$$

Square -176.

$$x=\frac{-\left(-176\right)±\sqrt{30976-4\times 100\left(-752\right)}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{-\left(-176\right)±\sqrt{30976-400\left(-752\right)}}{2\times 100}$$

Multiply -400 times -752.

$$x=\frac{-\left(-176\right)±\sqrt{30976+300800}}{2\times 100}$$

Add 30976 to 300800.

$$x=\frac{-\left(-176\right)±\sqrt{331776}}{2\times 100}$$

Take the square root of 331776.

$$x=\frac{-\left(-176\right)±576}{2\times 100}$$

The opposite of -176 is 176.

$$x=\frac{176±576}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{176±576}{200}$$

Now solve the equation x=\frac{176±576}{200} when ± is plus. Add 176 to 576.

$$x=\frac{752}{200}$$

Reduce the fraction \frac{752}{200} to lowest terms by extracting and canceling out 8.

$$x=\frac{94}{25}$$

Now solve the equation x=\frac{176±576}{200} when ± is minus. Subtract 576 from 176.

$$x=-\frac{400}{200}$$

Divide -400 by 200.

$$x=-2$$

The equation is now solved.

$$x=\frac{94}{25} x=-2$$

Steps for Completing the Square

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.

$$100x^{2}-176x-752=0$$

Add 752 to both sides of the equation.

$$100x^{2}-176x-752-\left(-752\right)=-\left(-752\right)$$

Subtracting -752 from itself leaves 0.

$$100x^{2}-176x=-\left(-752\right)$$

Subtract -752 from 0.

$$100x^{2}-176x=752$$

Divide both sides by 100.

$$\frac{100x^{2}-176x}{100}=\frac{752}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$x^{2}+\left(-\frac{176}{100}\right)x=\frac{752}{100}$$

Reduce the fraction \frac{-176}{100} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{44}{25}x=\frac{752}{100}$$

Reduce the fraction \frac{752}{100} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{44}{25}x=\frac{188}{25}$$

Divide -\frac{44}{25}, the coefficient of the x term, by 2 to get -\frac{22}{25}. Then add the square of -\frac{22}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{44}{25}x+\left(-\frac{22}{25}\right)^{2}=\frac{188}{25}+\left(-\frac{22}{25}\right)^{2}$$

Square -\frac{22}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{44}{25}x+\frac{484}{625}=\frac{188}{25}+\frac{484}{625}$$

Add \frac{188}{25} to \frac{484}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{44}{25}x+\frac{484}{625}=\frac{5184}{625}$$

Factor x^{2}-\frac{44}{25}x+\frac{484}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{22}{25}\right)^{2}=\frac{5184}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{22}{25}\right)^{2}}=\sqrt{\frac{5184}{625}}$$

Simplify.

$$x-\frac{22}{25}=\frac{72}{25} x-\frac{22}{25}=-\frac{72}{25}$$

Add \frac{22}{25} to both sides of the equation.

$$x=\frac{94}{25} x=-2$$