Find the vertex form of the following quadratic function:
2x^2-3x-2
The vertex form of a quadratic function is:
y=a\left(x-h\right)^{2}+k
Where \left(h,k\right) are the coordinates of the vertex.
Factor out the coefficient of x^{2}, in our problem 2:
2\left( x^{2}-\dfrac{3}{2}x-1 \right)
Complete a perfect square in the parenthesis:
2\left( x^{2}-2\cdot\dfrac{3}{4}x-1 \right)
2\left( x^{2}-2\cdot\dfrac{3}{4}x+\left( \dfrac{3}{4} \right)^{2}-\left( \dfrac{3}{4} \right)^{2}-1 \right)
Write the perfect square:
2\left[ \left( x-\dfrac{3}{4} \right)^{2}-\dfrac{9}{16}-1 \right]
Simplify the remaining part of the parenthesis:
2\left[ \left( x-\dfrac{3}{4} \right)^{2}-\dfrac{25}{16} \right]
Distribute the 2:
2\left( x-\dfrac{3}{4} \right)^{2}-\dfrac{25}{8}
2x^2-3x-2=2\left( x-\dfrac{3}{4} \right)^{2}-\dfrac{25}{8}
Find the vertex form of the following quadratic function:
5x^2-3x+7
The vertex form of a quadratic function is:
y=a\left(x-h\right)^{2}+k
Where \left(h,k\right) are the coordinates of the vertex.
Factor out the coefficient of x^{2}, in our problem 5:
5\left( x^{2}-\dfrac{3}{5}x+\dfrac{7}{5} \right)
Complete a perfect square in the parenthesis:
5\left( x^{2}-2\cdot\dfrac{3}{10}x+\dfrac{7}{5} \right)
5\left( x^{2}-2\cdot\dfrac{3}{10}x+\left( \dfrac{3}{10} \right)^{2}-\left( \dfrac{3}{10} \right)^{2}+\dfrac{7}{5} \right)
Write the perfect square:
5\left[ \left( x-\dfrac{3}{10} \right)^{2}-\dfrac{9}{100}+\dfrac{7}{5} \right]
Simplify the remaining part of the parenthesis:
5\left[ \left( x-\dfrac{3}{10} \right)^{2}+\dfrac{131}{100} \right]
Distribute the 5:
5\left( x-\dfrac{3}{10} \right)^{2}+\dfrac{131}{20}
5x^{2}-3x+7=5\left( x-\dfrac{3}{10} \right)^{2}+\dfrac{131}{20}
Find the vertex form of the following quadratic function:
3x^2+5x+1
The vertex form of a quadratic function is:
y=a\left(x-h\right)^{2}+k
Where \left(h,k\right) are the coordinates of the vertex.
Factor out the coefficient of x^{2}, in our problem 3:
3\left( x^{2}+\dfrac{5}{3}x+\dfrac{1}{3} \right)
Complete a perfect square in the parenthesis:
3\left( x^{2}+2\cdot\dfrac{5}{6}x+\dfrac{1}{3} \right)
3\left( x^{2}+2\cdot\dfrac{5}{6}x+\left( \dfrac{5}{6} \right)^{2}-\left( \dfrac{5}{6} \right)^{2}+\dfrac{1}{3} \right)
Write the perfect square:
3\left[ \left( x+\dfrac{5}{6} \right)^{2}-\dfrac{25}{36}+\dfrac{1}{3} \right]
Simplify the remaining part of the parenthesis:
3\left[ \left( x+\dfrac{5}{6} \right)^{2}-\dfrac{13}{36} \right]
Distribute the 3:
3\left( x+\dfrac{5}{6} \right)^{2}-\dfrac{13}{12}
3x^2+5x+1=3\left( x+\dfrac{5}{6} \right)^{2}-\dfrac{13}{12}
Find the vertex form of the following quadratic function:
2x^2+16x-15
The vertex form of a quadratic function is:
y=a\left(x-h\right)^{2}+k
Where \left(h,k\right) are the coordinates of the vertex.
Factor out the coefficient of x^{2}, in our problem 2:
2\left( x^{2}+8x-\dfrac{15}{2} \right)
Complete a perfect square in the parenthesis:
2\left( x^{2}+2\cdot4x-\dfrac{15}{2} \right)
2\left( x^{2}+2\cdot4x+\left( 4 \right)^{2}-\left( 4 \right)^{2}-\dfrac{15}{2} \right)
Write the perfect square:
2\left[ \left( x+4 \right)^{2}-16-\dfrac{15}{2} \right]
Simplify the remaining part of the parenthesis:
2\left[ \left( x+4 \right)^{2}-\dfrac{47}{2} \right]
Distribute the 2:
2\left( x+4 \right)^{2}-47
2x^{2}+16x-15=2\left( x+4 \right)^{2}-47
Find the standard form of the following quadratic function:
2\left(x-5\right)^{2}+4
In order to find the standard form of a quadratic function, simplify the given expression:
2\left(x-5\right)^{2}+4=2\left(x-5\right)\left(x-5\right)+4
2\left(x-5\right)^{2}+4=2\left(x^{2}-5x-5x+25\right)+4
2\left(x-5\right)^{2}+4=2\left(x^{2}-10x+25\right)+4
2\left(x-5\right)^{2}+4=2x^{2}-20x+50+4
2\left(x-5\right)^{2}+4=2x^{2}-20x+54
Find the standard form of the following quadratic function:
9\left( x-\dfrac{1}{3} \right)^{2}-7
In order to find the standard form of a quadratic function, simplify the given expression:
9\left( x-\dfrac{1}{3} \right)^{2}-7=9\left( x-\dfrac{1}{3} \right)\left( x-\dfrac{1}{3} \right)-7
9\left( x-\dfrac{1}{3} \right)^{2}-7=9\left( x^{2}-\dfrac{1}{3}x-\dfrac{1}{3}x+\dfrac{1}{9} \right)-7
9\left( x-\dfrac{1}{3} \right)^{2}-7=9\left( x^{2}-\dfrac{2}{3}x+\dfrac{1}{9} \right)-7
9\left( x-\dfrac{1}{3} \right)^{2}-7=9x^{2}-6x+1-7
9\left( x-\dfrac{1}{3} \right)^{2}-7=9x^{2}-6x-6
Find the standard form of the following quadratic function:
3\left( x+\dfrac{1}{6} \right)^{2}-\dfrac{133}{12}
In order to find the standard form of a quadratic function, simplify the given expression:
3\left( x+\dfrac{1}{6} \right)^{2}-\dfrac{133}{12}=3\left( x+\dfrac{1}{6} \right)\left( x+\dfrac{1}{6} \right)-\dfrac{133}{12}
3\left( x+\dfrac{1}{6} \right)^{2}-\dfrac{133}{12}=3\left( x^{2}+\dfrac{1}{6}x+\dfrac{1}{6}x+\dfrac{1}{36} \right)-\dfrac{133}{12}
3\left( x+\dfrac{1}{6} \right)^{2}-\dfrac{133}{12}=3\left( x^{2}+\dfrac{1}{3}x+\dfrac{1}{36} \right)-\dfrac{133}{12}
3\left( x+\dfrac{1}{6} \right)^{2}-\dfrac{133}{12}=3x^{2}+x+\dfrac{1}{12}-\dfrac{133}{12}
3\left( x+\dfrac{1}{6} \right)^{2}-\dfrac{133}{12}=3x^{2}+x-11