Solve -5x^2-25x+70

$$- 5 x ^ { 2 } - 25 x + 70$$

Factor

$$5\left(2-x\right)\left(x+7\right)$$

Solution Steps

Factor out 5.

$$5\left(-x^{2}-5x+14\right)$$

Consider -x^{2}-5x+14. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+14. To find a and b, set up a system to be solved.

$$a+b=-5 ab=-14=-14$$

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 -14.

$$1,-14 2,-7$$

Calculate the sum for each pair.

$$1-14=-13 2-7=-5$$

The solution is the pair that gives sum -5.

$$a=2 b=-7$$

Rewrite -x^{2}-5x+14 as \left(-x^{2}+2x\right)+\left(-7x+14\right).

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

Factor out x in the first and 7 in the second group.

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

Factor out common term -x+2 by using distributive property.

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

Rewrite the complete factored expression.

$$5\left(-x+2\right)\left(x+7\right)$$

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.

$$-5x^{2}-25x+70=0$$

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.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\left(-5\right)\times 70}}{2\left(-5\right)}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\left(-5\right)\times 70}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-25\right)±\sqrt{625+20\times 70}}{2\left(-5\right)}$$

Multiply 20 times 70.

$$x=\frac{-\left(-25\right)±\sqrt{625+1400}}{2\left(-5\right)}$$

Add 625 to 1400.

$$x=\frac{-\left(-25\right)±\sqrt{2025}}{2\left(-5\right)}$$

Take the square root of 2025.

$$x=\frac{-\left(-25\right)±45}{2\left(-5\right)}$$

The opposite of -25 is 25.

$$x=\frac{25±45}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{25±45}{-10}$$

Now solve the equation x=\frac{25±45}{-10} when ± is plus. Add 25 to 45.

$$x=\frac{70}{-10}$$

Divide 70 by -10.

$$x=-7$$

Now solve the equation x=\frac{25±45}{-10} when ± is minus. Subtract 45 from 25.

$$x=-\frac{20}{-10}$$

Divide -20 by -10.

$$x=2$$

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

$$-5x^{2}-25x+70=-5\left(x-\left(-7\right)\right)\left(x-2\right)$$

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

$$-5x^{2}-25x+70=-5\left(x+7\right)\left(x-2\right)$$

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.

$$x ^ 2 +5x -14 = 0$$

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

$$r + s = -5 rs = -14$$

Two numbers r and s sum up to -5 exactly when the average of the two numbers is \frac{1}{2}*-5 = -\frac{5}{2}. 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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -14

$$(-\frac{5}{2} - u) (-\frac{5}{2} + u) = -14$$

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

$$\frac{25}{4} - u^2 = -14$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = -14-\frac{25}{4} = -\frac{81}{4}$$

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

$$u^2 = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}$$

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

$$r =-\frac{5}{2} - \frac{9}{2} = -7 s = -\frac{5}{2} + \frac{9}{2} = 2$$

Evaluate

$$5\left(2-x\right)\left(x+7\right)$$