Solve y=x^(2)-5x-14

Question

Function

Find the vertex

Find the axis of symmetry

Rewrite in vertex form

(\frac{5}{2}, - \frac{81}{4})

Evaluate

y = x^{2} - 5 x - 14

Find the x -coordinate of the vertex by substituting a= 1 and b= - 5 into x = - \frac{b}{2 a}

x = - \frac{- 5}{2 \times 1}

Solve the equation for x

x = \frac{5}{2}

Find the y-coordinate of the vertex by evaluating the function for x = \frac{5}{2}

y = (\frac{5}{2})^{2} - 5 \times \frac{5}{2} - 14

Calculate

More Steps

Evaluate

(\frac{5}{2})^{2} - 5 \times \frac{5}{2} - 14

Multiply the numbers

More Steps

Evaluate

- 5 \times \frac{5}{2}

Multiply the numbers

- \frac{5 \times 5}{2}

Multiply the numbers

- \frac{25}{2}

(\frac{5}{2})^{2} - \frac{25}{2} - 14

Evaluate the power

\frac{25}{4} - \frac{25}{2} - 14

Reduce fractions to a common denominator

\frac{25}{4} - \frac{25 \times 2}{2 \times 2} - \frac{14 \times 2 \times 2}{2 \times 2}

Multiply the numbers

\frac{25}{4} - \frac{25 \times 2}{4} - \frac{14 \times 2 \times 2}{2 \times 2}

Multiply the numbers

\frac{25}{4} - \frac{25 \times 2}{4} - \frac{14 \times 2 \times 2}{4}

Write all numerators above the common denominator

\frac{25 - 25 \times 2 - 14 \times 2 \times 2}{4}

Multiply the numbers

\frac{25 - 50 - 14 \times 2 \times 2}{4}

Multiply the terms

More Steps

Evaluate

14 \times 2 \times 2

Multiply the terms

28 \times 2

Multiply the numbers

56

\frac{25 - 50 - 56}{4}

Subtract the numbers

\frac{- 81}{4}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{81}{4}

y = - \frac{81}{4}

Solution

(\frac{5}{2}, - \frac{81}{4})

Show Solution

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

Evaluate

y = x^{2} - 5 x - 14

To test if the graph of y = x^{2} - 5 x - 14 is symmetry with respect to the origin,substitute -x for x and -y for y

- y = (- x)^{2} - 5 (- x) - 14

Simplify

More Steps

Evaluate

(- x)^{2} - 5 (- x) - 14

Multiply the numbers

(- x)^{2} + 5 x - 14

Rewrite the expression

x^{2} + 5 x - 14

- y = x^{2} + 5 x - 14

Change the signs both sides

y = - x^{2} - 5 x + 14

Solution

Not symmetry with respect to the origin

Show Solution

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

(x - \frac{5}{2})^{2} = y + \frac{81}{4}

Evaluate

y = x^{2} - 5 x - 14

Swap the sides of the equation

x^{2} - 5 x - 14 = y

Move the constant to the right-hand side and change its sign

x^{2} - 5 x = y - (- 14)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

x^{2} - 5 x = y + 14

To complete the square, the same value needs to be added to both sides

x^{2} - 5 x + \frac{25}{4} = y + 14 + \frac{25}{4}

Use a^{2} - 2 ab + b^{2} = (a - b)^{2} to factor the expression

(x - \frac{5}{2})^{2} = y + 14 + \frac{25}{4}

Solution

More Steps

Evaluate

14 + \frac{25}{4}

Reduce fractions to a common denominator

\frac{14 \times 4}{4} + \frac{25}{4}

Write all numerators above the common denominator

\frac{14 \times 4 + 25}{4}

Multiply the numbers

\frac{56 + 25}{4}

Add the numbers

\frac{81}{4}

(x - \frac{5}{2})^{2} = y + \frac{81}{4}

Show Solution

Solve the equation

$Solve for x$

x = \frac{81 + 4 y + 5}{2} x = \frac{- 81 + 4 y + 5}{2}

Evaluate

y = x^{2} - 5 x - 14

Swap the sides of the equation

x^{2} - 5 x - 14 = y

Move the expression to the left side

x^{2} - 5 x - 14 - y = 0

Move the constant to the right side

x^{2} - 5 x = 0 - (- 14 - y)

Add the terms

x^{2} - 5 x = 14 + y

Add the same value to both sides

x^{2} - 5 x + \frac{25}{4} = 14 + y + \frac{25}{4}

Evaluate

x^{2} - 5 x + \frac{25}{4} = \frac{81 + 4 y}{4}

Evaluate

(x - \frac{5}{2})^{2} = \frac{81 + 4 y}{4}

Take the root of both sides of the equation and remember to use both positive and negative roots

x - \frac{5}{2} = \pm \frac{81 + 4 y}{4}

Simplify the expression

More Steps

Evaluate

\frac{81 + 4 y}{4}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{81 + 4 y}{4}

Simplify the radical expression

More Steps

Evaluate

4

Write the number in exponential form with the base of 2

2^{2}

Reduce the index of the radical and exponent with 2

2

\frac{81 + 4 y}{2}

x - \frac{5}{2} = \pm \frac{81 + 4 y}{2}

Separate the equation into 2 possible cases

x - \frac{5}{2} = \frac{81 + 4 y}{2} x - \frac{5}{2} = - \frac{81 + 4 y}{2}

Calculate

More Steps

Evaluate

x - \frac{5}{2} = \frac{81 + 4 y}{2}

Move the constant to the right-hand side and change its sign

x = \frac{81 + 4 y}{2} + \frac{5}{2}

Write all numerators above the common denominator

x = \frac{81 + 4 y + 5}{2}

x = \frac{81 + 4 y + 5}{2} x - \frac{5}{2} = - \frac{81 + 4 y}{2}

Solution

More Steps

Evaluate

x - \frac{5}{2} = - \frac{81 + 4 y}{2}

Move the constant to the right-hand side and change its sign

x = - \frac{81 + 4 y}{2} + \frac{5}{2}

Write all numerators above the common denominator

x = \frac{- 81 + 4 y + 5}{2}

x = \frac{81 + 4 y + 5}{2} x = \frac{- 81 + 4 y + 5}{2}

Show Solution

Rewrite the equation

Rewrite in polar form

r = \frac{sin ( θ ) + 5 cos ( θ ) - 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{2 cos ^{2} ( θ )} r = \frac{sin ( θ ) + 5 cos ( θ ) + 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{2 cos ^{2} ( θ )}

Evaluate

y = x^{2} - 5 x - 14

Move the expression to the left side

y - x^{2} + 5 x = - 14

To convert the equation to polar coordinates,substitute r cos (θ) for x and r sin (θ) for y

sin (θ) \times r - (cos (θ) \times r)^{2} + 5 cos (θ) \times r = - 14

Factor the expression

- cos^{2} (θ) \times r^{2} + (sin (θ) + 5 cos (θ)) r = - 14

Subtract the terms

- cos^{2} (θ) \times r^{2} + (sin (θ) + 5 cos (θ)) r - (- 14) = - 14 - (- 14)

Evaluate

- cos^{2} (θ) \times r^{2} + (sin (θ) + 5 cos (θ)) r + 14 = 0

Solve using the quadratic formula

r = \frac{- sin ( θ ) - 5 cos ( θ ) \pm ( sin ( θ ) + 5 cos ( θ ) ) ^{2} - 4 ( - cos ^{2} ( θ ) ) \times 14}{- 2 cos ^{2} ( θ )}

Simplify

r = \frac{- sin ( θ ) - 5 cos ( θ ) \pm 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{- 2 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- sin ( θ ) - 5 cos ( θ ) + 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{- 2 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 5 cos ( θ ) - 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{- 2 cos ^{2} ( θ )}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

r = \frac{sin ( θ ) + 5 cos ( θ ) - 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{2 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 5 cos ( θ ) - 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{- 2 cos ^{2} ( θ )}

Solution

r = \frac{sin ( θ ) + 5 cos ( θ ) - 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{2 cos ^{2} ( θ )} r = \frac{sin ( θ ) + 5 cos ( θ ) + 1 + 80 cos ^{2} ( θ ) + 5 sin ( 2 θ )}{2 cos ^{2} ( θ )}

Show Solution

Y=x^(2) 5x 14

Function

Find the vertex

Find the axis of symmetry

Rewrite in vertex form

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

Solve the equation

$Solve for x$

Rewrite the equation

Rewrite in polar form

Graph