Solve x^2+x-30 | Microsoft Math Solver (2024)

a+b=1 ab=1\left(-30\right)=-30

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

-1,30 -2,15 -3,10 -5,6

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 -30.

-1+30=29 -2+15=13 -3+10=7 -5+6=1

Calculate the sum for each pair.

a=-5 b=6

The solution is the pair that gives sum 1.

\left(x^{2}-5x\right)+\left(6x-30\right)

Rewrite x^{2}+x-30 as \left(x^{2}-5x\right)+\left(6x-30\right).

x\left(x-5\right)+6\left(x-5\right)

Factor out x in the first and 6 in the second group.

\left(x-5\right)\left(x+6\right)

Factor out common term x-5 by using distributive property.

x^{2}+x-30=0

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.

x=\frac{-1±\sqrt{1^{2}-4\left(-30\right)}}{2}

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{-1±\sqrt{1-4\left(-30\right)}}{2}

Square 1.

x=\frac{-1±\sqrt{1+120}}{2}

Multiply -4 times -30.

x=\frac{-1±\sqrt{121}}{2}

Add 1 to 120.

x=\frac{-1±11}{2}

Take the square root of 121.

x=\frac{10}{2}

Now solve the equation x=\frac{-1±11}{2} when ± is plus. Add -1 to 11.

x=5

Divide 10 by 2.

x=-\frac{12}{2}

Now solve the equation x=\frac{-1±11}{2} when ± is minus. Subtract 11 from -1.

x=-6

Divide -12 by 2.

x^{2}+x-30=\left(x-5\right)\left(x-\left(-6\right)\right)

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and -6 for x_{2}.

x^{2}+x-30=\left(x-5\right)\left(x+6\right)

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

x ^ 2 +1x -30 = 0

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.

r + s = -1 rs = -30

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 = -\frac{1}{2} - u s = -\frac{1}{2} + u

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

(-\frac{1}{2} - u) (-\frac{1}{2} + u) = -30

To solve for unknown quantity u, substitute these in the product equation rs = -30

\frac{1}{4} - u^2 = -30

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

-u^2 = -30-\frac{1}{4} = -\frac{121}{4}

Simplify the expression by subtracting \frac{1}{4} on both sides

u^2 = \frac{121}{4} u = \pm\sqrt{\frac{121}{4}} = \pm \frac{11}{2}

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

r =-\frac{1}{2} - \frac{11}{2} = -6 s = -\frac{1}{2} + \frac{11}{2} = 5

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