Solve -7x^2-4x+20

$$- 7 x ^ { 2 } - 4 x + 20$$

Factor

$$-\left(7x-10\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -7x^{2}+ax+bx+20. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-7\times 20=-140$$

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 -140.

$$1,-140 2,-70 4,-35 5,-28 7,-20 10,-14$$

Calculate the sum for each pair.

$$1-140=-139 2-70=-68 4-35=-31 5-28=-23 7-20=-13 10-14=-4$$

The solution is the pair that gives sum -4.

$$a=10 b=-14$$

Rewrite -7x^{2}-4x+20 as \left(-7x^{2}+10x\right)+\left(-14x+20\right).

$$\left(-7x^{2}+10x\right)+\left(-14x+20\right)$$

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

$$-x\left(7x-10\right)-2\left(7x-10\right)$$

Factor out common term 7x-10 by using distributive property.

$$\left(7x-10\right)\left(-x-2\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-7x^{2}-4x+20=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-7\right)\times 20}}{2\left(-7\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-7\right)\times 20}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-4\right)±\sqrt{16+28\times 20}}{2\left(-7\right)}$$

Multiply 28 times 20.

$$x=\frac{-\left(-4\right)±\sqrt{16+560}}{2\left(-7\right)}$$

Add 16 to 560.

$$x=\frac{-\left(-4\right)±\sqrt{576}}{2\left(-7\right)}$$

Take the square root of 576.

$$x=\frac{-\left(-4\right)±24}{2\left(-7\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±24}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{4±24}{-14}$$

Now solve the equation x=\frac{4±24}{-14} when ± is plus. Add 4 to 24.

$$x=\frac{28}{-14}$$

Divide 28 by -14.

$$x=-2$$

Now solve the equation x=\frac{4±24}{-14} when ± is minus. Subtract 24 from 4.

$$x=-\frac{20}{-14}$$

Reduce the fraction \frac{-20}{-14} to lowest terms by extracting and canceling out 2.

$$x=\frac{10}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and \frac{10}{7} for x_{2}.

$$-7x^{2}-4x+20=-7\left(x-\left(-2\right)\right)\left(x-\frac{10}{7}\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-7x^{2}-4x+20=-7\left(x+2\right)\left(x-\frac{10}{7}\right)$$

Subtract \frac{10}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-7x^{2}-4x+20=-7\left(x+2\right)\times \frac{-7x+10}{-7}$$

Cancel out 7, the greatest common factor in -7 and 7.

$$-7x^{2}-4x+20=\left(x+2\right)\left(-7x+10\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +\frac{4}{7}x -\frac{20}{7} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -\frac{4}{7} rs = -\frac{20}{7}$$

Two numbers r and s sum up to -\frac{4}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{4}{7} = -\frac{2}{7}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{2}{7} - u s = -\frac{2}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{20}{7}

$$(-\frac{2}{7} - u) (-\frac{2}{7} + u) = -\frac{20}{7}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{4}{49} - u^2 = -\frac{20}{7}$$

Simplify the expression by subtracting \frac{4}{49} on both sides

$$-u^2 = -\frac{20}{7}-\frac{4}{49} = -\frac{144}{49}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{144}{49} u = \pm\sqrt{\frac{144}{49}} = \pm \frac{12}{7}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{2}{7} - \frac{12}{7} = -2 s = -\frac{2}{7} + \frac{12}{7} = 1.429$$

Evaluate

$$-\left(7x-10\right)\left(x+2\right)$$