Solve 5p^2-0p+10=0

$$5 p ^ { 2 } - 0 p + 10 = 0$$

Solve for p

$$p=-\sqrt{2}i\approx -0-1.414213562i$$

$$p=\sqrt{2}i\approx 1.414213562i$$

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.

$$5p^{2}+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 0 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{0±\sqrt{0^{2}-4\times 5\times 10}}{2\times 5}$$

Square 0.

$$p=\frac{0±\sqrt{-4\times 5\times 10}}{2\times 5}$$

Multiply -4 times 5.

$$p=\frac{0±\sqrt{-20\times 10}}{2\times 5}$$

Multiply -20 times 10.

$$p=\frac{0±\sqrt{-200}}{2\times 5}$$

Take the square root of -200.

$$p=\frac{0±10\sqrt{2}i}{2\times 5}$$

Multiply 2 times 5.

$$p=\frac{0±10\sqrt{2}i}{10}$$

Now solve the equation p=\frac{0±10\sqrt{2}i}{10} when ± is plus.

$$p=\sqrt{2}i$$

Now solve the equation p=\frac{0±10\sqrt{2}i}{10} when ± is minus.

$$p=-\sqrt{2}i$$

The equation is now solved.

$$p=\sqrt{2}i p=-\sqrt{2}i$$

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.This is achieved by dividing both sides of the equation by 5

$$x ^ 2 +0x +2 = 0$$

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

$$r + s = 0 rs = 2$$

Two numbers r and s sum up to 0 exactly when the average of the two numbers is \frac{1}{2}*0 = 0. 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>

$$r = 0 - u s = 0 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 2

$$(0 - u) (0 + u) = 2$$

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

$$0 - u^2 = 2$$

Simplify the expression by subtracting 0 on both sides

$$-u^2 = 2-0 = 2$$

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

$$u^2 = -2 u = \pm\sqrt{-2} = \pm \sqrt{2}i$$

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

$$r =0 - \sqrt{2}i s = 0 + \sqrt{2}i$$