Solve for x

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Steps Using the Quadratic Formula
Steps for Completing the Square
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a+b=-13 ab=42
To solve the equation, factor x^{2}-13x+42 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
-1,-42 -2,-21 -3,-14 -6,-7
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 42.
-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13
Calculate the sum for each pair.
a=-7 b=-6
The solution is the pair that gives sum -13.
\left(x-7\right)\left(x-6\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=7 x=6
To find equation solutions, solve x-7=0 and x-6=0.
a+b=-13 ab=1\times 42=42
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+42. To find a and b, set up a system to be solved.
-1,-42 -2,-21 -3,-14 -6,-7
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 42.
-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13
Calculate the sum for each pair.
a=-7 b=-6
The solution is the pair that gives sum -13.
\left(x^{2}-7x\right)+\left(-6x+42\right)
Rewrite x^{2}-13x+42 as \left(x^{2}-7x\right)+\left(-6x+42\right).
x\left(x-7\right)-6\left(x-7\right)
Factor out x in the first and -6 in the second group.
\left(x-7\right)\left(x-6\right)
Factor out common term x-7 by using distributive property.
x=7 x=6
To find equation solutions, solve x-7=0 and x-6=0.
x^{2}-13x+42=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(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 42}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -13 for b, and 42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-13\right)±\sqrt{169-4\times 42}}{2}
Square -13.
x=\frac{-\left(-13\right)±\sqrt{169-168}}{2}
Multiply -4 times 42.
x=\frac{-\left(-13\right)±\sqrt{1}}{2}
Add 169 to -168.
x=\frac{-\left(-13\right)±1}{2}
Take the square root of 1.
x=\frac{13±1}{2}
The opposite of -13 is 13.
x=\frac{14}{2}
Now solve the equation x=\frac{13±1}{2} when ± is plus. Add 13 to 1.
x=7
Divide 14 by 2.
x=\frac{12}{2}
Now solve the equation x=\frac{13±1}{2} when ± is minus. Subtract 1 from 13.
x=6
Divide 12 by 2.
x=7 x=6
The equation is now solved.
x^{2}-13x+42=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
x^{2}-13x+42-42=-42
Subtract 42 from both sides of the equation.
x^{2}-13x=-42
Subtracting 42 from itself leaves 0.
x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-42+\left(-\frac{13}{2}\right)^{2}
Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}=-6.5. Then add the square of -\frac{13}{2}=-6.5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-13x+\frac{169}{4}=-42+\frac{169}{4}
Square -\frac{13}{2}=-6.5 by squaring both the numerator and the denominator of the fraction.
x^{2}-13x+\frac{169}{4}=\frac{1}{4}
Add -42 to \frac{169}{4}=42.25.
\left(x-\frac{13}{2}\right)^{2}=\frac{1}{4}
Factor x^{2}-13x+\frac{169}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{1}{4}}
Take the square root of both sides of the equation.
x-\frac{13}{2}=\frac{1}{2} x-\frac{13}{2}=-\frac{1}{2}
Simplify.
x=7 x=6
Add \frac{13}{2}=6.5 to both sides of the equation.
x ^ 2 -13x +42 = 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 = 13 rs = 42
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{13}{2} - u s = \frac{13}{2} + u
Two numbers r and s sum up to 13 exactly when the average of the two numbers is \frac{1}{2}*13 = \frac{13}{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{13}{2} - u) (\frac{13}{2} + u) = 42
To solve for unknown quantity u, substitute these in the product equation rs = 42
\frac{169}{4} - u^2 = 42
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 42-\frac{169}{4} = -\frac{1}{4}
Simplify the expression by subtracting \frac{169}{4} on both sides
u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{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{13}{2} - \frac{1}{2} = 6 s = \frac{13}{2} + \frac{1}{2} = 7
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.