Solve p^2+24p-25

$$p ^ { 2 } + 24 p - 25$$

Factor

$$\left(p-1\right)\left(p+25\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as p^{2}+ap+bp-25. To find a and b, set up a system to be solved.

$$a+b=24 ab=1\left(-25\right)=-25$$

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 -25.

$$-1,25 -5,5$$

Calculate the sum for each pair.

$$-1+25=24 -5+5=0$$

The solution is the pair that gives sum 24.

$$a=-1 b=25$$

Rewrite p^{2}+24p-25 as \left(p^{2}-p\right)+\left(25p-25\right).

$$\left(p^{2}-p\right)+\left(25p-25\right)$$

Factor out p in the first and 25 in the second group.

$$p\left(p-1\right)+25\left(p-1\right)$$

Factor out common term p-1 by using distributive property.

$$\left(p-1\right)\left(p+25\right)$$

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.

$$p^{2}+24p-25=0$$

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.

$$p=\frac{-24±\sqrt{24^{2}-4\left(-25\right)}}{2}$$

Square 24.

$$p=\frac{-24±\sqrt{576-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$p=\frac{-24±\sqrt{576+100}}{2}$$

Add 576 to 100.

$$p=\frac{-24±\sqrt{676}}{2}$$

Take the square root of 676.

$$p=\frac{-24±26}{2}$$

Now solve the equation p=\frac{-24±26}{2} when ± is plus. Add -24 to 26.

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

Divide 2 by 2.

$$p=1$$

Now solve the equation p=\frac{-24±26}{2} when ± is minus. Subtract 26 from -24.

$$p=-\frac{50}{2}$$

Divide -50 by 2.

$$p=-25$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -25 for x_{2}.

$$p^{2}+24p-25=\left(p-1\right)\left(p-\left(-25\right)\right)$$

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

$$p^{2}+24p-25=\left(p-1\right)\left(p+25\right)$$

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.

$$x ^ 2 +24x -25 = 0$$

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

$$r + s = -24 rs = -25$$

Two numbers r and s sum up to -24 exactly when the average of the two numbers is \frac{1}{2}*-24 = -12. 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>

$$r = -12 - u s = -12 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -25

$$(-12 - u) (-12 + u) = -25$$

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

$$144 - u^2 = -25$$

Simplify the expression by subtracting 144 on both sides

$$-u^2 = -25-144 = -169$$

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

$$u^2 = 169 u = \pm\sqrt{169} = \pm 13$$

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

$$r =-12 - 13 = -25 s = -12 + 13 = 1$$

Evaluate

$$\left(p-1\right)\left(p+25\right)$$