Solve -4a^2-14a+20

$$- 4 a ^ { 2 } - 14 a + 20$$

Factor

$$-4\left(a-\frac{-\sqrt{129}-7}{4}\right)\left(a-\frac{\sqrt{129}-7}{4}\right)$$

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.

$$-4a^{2}-14a+20=0$$

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.

$$a=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-4\right)\times 20}}{2\left(-4\right)}$$

Square -14.

$$a=\frac{-\left(-14\right)±\sqrt{196-4\left(-4\right)\times 20}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$a=\frac{-\left(-14\right)±\sqrt{196+16\times 20}}{2\left(-4\right)}$$

Multiply 16 times 20.

$$a=\frac{-\left(-14\right)±\sqrt{196+320}}{2\left(-4\right)}$$

Add 196 to 320.

$$a=\frac{-\left(-14\right)±\sqrt{516}}{2\left(-4\right)}$$

Take the square root of 516.

$$a=\frac{-\left(-14\right)±2\sqrt{129}}{2\left(-4\right)}$$

The opposite of -14 is 14.

$$a=\frac{14±2\sqrt{129}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$a=\frac{14±2\sqrt{129}}{-8}$$

Now solve the equation a=\frac{14±2\sqrt{129}}{-8} when ± is plus. Add 14 to 2\sqrt{129}.

$$a=\frac{2\sqrt{129}+14}{-8}$$

Divide 14+2\sqrt{129} by -8.

$$a=\frac{-\sqrt{129}-7}{4}$$

Now solve the equation a=\frac{14±2\sqrt{129}}{-8} when ± is minus. Subtract 2\sqrt{129} from 14.

$$a=\frac{14-2\sqrt{129}}{-8}$$

Divide 14-2\sqrt{129} by -8.

$$a=\frac{\sqrt{129}-7}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7-\sqrt{129}}{4} for x_{1} and \frac{-7+\sqrt{129}}{4} for x_{2}.

$$-4a^{2}-14a+20=-4\left(a-\frac{-\sqrt{129}-7}{4}\right)\left(a-\frac{\sqrt{129}-7}{4}\right)$$

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.

$$x ^ 2 +\frac{7}{2}x -5 = 0$$

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

$$r + s = -\frac{7}{2} rs = -5$$

Two numbers r and s sum up to -\frac{7}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{2} = -\frac{7}{4}. 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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -5

$$(-\frac{7}{4} - u) (-\frac{7}{4} + u) = -5$$

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

$$\frac{49}{16} - u^2 = -5$$

Simplify the expression by subtracting \frac{49}{16} on both sides

$$-u^2 = -5-\frac{49}{16} = -\frac{129}{16}$$

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

$$u^2 = \frac{129}{16} u = \pm\sqrt{\frac{129}{16}} = \pm \frac{\sqrt{129}}{4}$$

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

$$r =-\frac{7}{4} - \frac{\sqrt{129}}{4} = -4.589 s = -\frac{7}{4} + \frac{\sqrt{129}}{4} = 1.089$$

Evaluate

$$20-14a-4a^{2}$$