Solve -p^2+3p+70

$$- p ^ { 2 } + 3 p + 70$$

Factor

$$-\left(p-10\right)\left(p+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -p^{2}+ap+bp+70. To find a and b, set up a system to be solved.

$$a+b=3 ab=-70=-70$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -70.

$$-1,70 -2,35 -5,14 -7,10$$

Calculate the sum for each pair.

$$-1+70=69 -2+35=33 -5+14=9 -7+10=3$$

The solution is the pair that gives sum 3.

$$a=10 b=-7$$

Rewrite -p^{2}+3p+70 as \left(-p^{2}+10p\right)+\left(-7p+70\right).

$$\left(-p^{2}+10p\right)+\left(-7p+70\right)$$

Factor out -p in the first and -7 in the second group.

$$-p\left(p-10\right)-7\left(p-10\right)$$

Factor out common term p-10 by using distributive property.

$$\left(p-10\right)\left(-p-7\right)$$

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.

$$-p^{2}+3p+70=0$$

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.

$$p=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 70}}{2\left(-1\right)}$$

Square 3.

$$p=\frac{-3±\sqrt{9-4\left(-1\right)\times 70}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$p=\frac{-3±\sqrt{9+4\times 70}}{2\left(-1\right)}$$

Multiply 4 times 70.

$$p=\frac{-3±\sqrt{9+280}}{2\left(-1\right)}$$

Add 9 to 280.

$$p=\frac{-3±\sqrt{289}}{2\left(-1\right)}$$

Take the square root of 289.

$$p=\frac{-3±17}{2\left(-1\right)}$$

Multiply 2 times -1.

$$p=\frac{-3±17}{-2}$$

Now solve the equation p=\frac{-3±17}{-2} when ± is plus. Add -3 to 17.

$$p=\frac{14}{-2}$$

Divide 14 by -2.

$$p=-7$$

Now solve the equation p=\frac{-3±17}{-2} when ± is minus. Subtract 17 from -3.

$$p=-\frac{20}{-2}$$

Divide -20 by -2.

$$p=10$$

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 10 for x_{2}.

$$-p^{2}+3p+70=-\left(p-\left(-7\right)\right)\left(p-10\right)$$

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

$$-p^{2}+3p+70=-\left(p+7\right)\left(p-10\right)$$

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.

$$x ^ 2 -3x -70 = 0$$

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

$$r + s = 3 rs = -70$$

Two numbers r and s sum up to 3 exactly when the average of the two numbers is \frac{1}{2}*3 = \frac{3}{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>

$$r = \frac{3}{2} - u s = \frac{3}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -70

$$(\frac{3}{2} - u) (\frac{3}{2} + u) = -70$$

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

$$\frac{9}{4} - u^2 = -70$$

Simplify the expression by subtracting \frac{9}{4} on both sides

$$-u^2 = -70-\frac{9}{4} = -\frac{289}{4}$$

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

$$u^2 = \frac{289}{4} u = \pm\sqrt{\frac{289}{4}} = \pm \frac{17}{2}$$

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

$$r =\frac{3}{2} - \frac{17}{2} = -7 s = \frac{3}{2} + \frac{17}{2} = 10$$

Evaluate

$$-\left(p-10\right)\left(p+7\right)$$