Solve for k

k = -\frac{7}{2} = -3\frac{1}{2} = -3.5<br/>k=-1

$k=−27 =−321 =−3.5$

$k=−1$

$k=−1$

Steps Using Factoring By Grouping

Steps Using the Quadratic Formula

Steps for Completing the Square

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2k^{2}+9k+7=0

Add 7 to both sides.

a+b=9 ab=2\times 7=14

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2k^{2}+ak+bk+7. To find a and b, set up a system to be solved.

1,14 2,7

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 14.

1+14=15 2+7=9

Calculate the sum for each pair.

a=2 b=7

The solution is the pair that gives sum 9.

\left(2k^{2}+2k\right)+\left(7k+7\right)

Rewrite 2k^{2}+9k+7 as \left(2k^{2}+2k\right)+\left(7k+7\right).

2k\left(k+1\right)+7\left(k+1\right)

Factor out 2k in the first and 7 in the second group.

\left(k+1\right)\left(2k+7\right)

Factor out common term k+1 by using distributive property.

k=-1 k=-\frac{7}{2}

To find equation solutions, solve k+1=0 and 2k+7=0.

2k^{2}+9k=-7

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.

2k^{2}+9k-\left(-7\right)=-7-\left(-7\right)

Add 7 to both sides of the equation.

2k^{2}+9k-\left(-7\right)=0

Subtracting -7 from itself leaves 0.

2k^{2}+9k+7=0

Subtract -7 from 0.

k=\frac{-9±\sqrt{9^{2}-4\times 2\times 7}}{2\times 2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 9 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

k=\frac{-9±\sqrt{81-4\times 2\times 7}}{2\times 2}

Square 9.

k=\frac{-9±\sqrt{81-8\times 7}}{2\times 2}

Multiply -4 times 2.

k=\frac{-9±\sqrt{81-56}}{2\times 2}

Multiply -8 times 7.

k=\frac{-9±\sqrt{25}}{2\times 2}

Add 81 to -56.

k=\frac{-9±5}{2\times 2}

Take the square root of 25.

k=\frac{-9±5}{4}

Multiply 2 times 2.

k=\frac{-4}{4}

Now solve the equation k=\frac{-9±5}{4} when ± is plus. Add -9 to 5.

k=-1

Divide -4 by 4.

k=\frac{-14}{4}

Now solve the equation k=\frac{-9±5}{4} when ± is minus. Subtract 5 from -9.

k=-\frac{7}{2}

Reduce the fraction \frac{-14}{4}=-3.5 to lowest terms by extracting and canceling out 2.

k=-1 k=-\frac{7}{2}

The equation is now solved.

2k^{2}+9k=-7

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.

\frac{2k^{2}+9k}{2}=\frac{-7}{2}

Divide both sides by 2.

k^{2}+\frac{9}{2}k=\frac{-7}{2}

Dividing by 2 undoes the multiplication by 2.

k^{2}+\frac{9}{2}k=-\frac{7}{2}

Divide -7 by 2.

k^{2}+\frac{9}{2}k+\left(\frac{9}{4}\right)^{2}=-\frac{7}{2}+\left(\frac{9}{4}\right)^{2}

Divide \frac{9}{2}=4.5, the coefficient of the x term, by 2 to get \frac{9}{4}=2.25. Then add the square of \frac{9}{4}=2.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

k^{2}+\frac{9}{2}k+\frac{81}{16}=-\frac{7}{2}+\frac{81}{16}

Square \frac{9}{4}=2.25 by squaring both the numerator and the denominator of the fraction.

k^{2}+\frac{9}{2}k+\frac{81}{16}=\frac{25}{16}

Add -\frac{7}{2}=-3.5 to \frac{81}{16}=5.0625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

\left(k+\frac{9}{4}\right)^{2}=\frac{25}{16}

Factor k^{2}+\frac{9}{2}k+\frac{81}{16}. 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(k+\frac{9}{4}\right)^{2}}=\sqrt{\frac{25}{16}}

Take the square root of both sides of the equation.

k+\frac{9}{4}=\frac{5}{4} k+\frac{9}{4}=-\frac{5}{4}

Simplify.

k=-1 k=-\frac{7}{2}

Subtract \frac{9}{4}=2.25 from both sides of the equation.

Examples

Quadratic equation

{ x } ^ { 2 } - 4 x - 5 = 0

$x_{2}−4x−5=0$

Trigonometry

4 \sin \theta \cos \theta = 2 \sin \theta

$4sinθcosθ=2sinθ$

Linear equation

y = 3x + 4

$y=3x+4$

Arithmetic

699 * 533

$699∗533$

Matrix

\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { - 1 } & { 1 } & { 5 } \end{array} \right]

$[25 34 ][2−1 01 35 ]$

Simultaneous equation

\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.

${8x+2y=467x+3y=47 $

Differentiation

\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }

$dxd (x−5)(3x_{2}−2) $

Integration

\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x

$∫_{0}xe_{−x_{2}}dx$

Limits

\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}

$x→−3lim x_{2}+2x−3x_{2}−9 $