1.05 Quadratic formula

Below is the result after using the quadratic formula to solve an equation:

m = \dfrac{- \left( - 10 \right) \pm \sqrt{ - 1 }}{8}

What can be concluded about the solutions of the equation?

How many real solutions do the following equations have?

For the following equations:

Consider the equation x^{2} + 22 x + 121 = 0.

State whether each of the following equations have any real solutions:

For the following equations:

Consider the equation x^{2} - 8 x - 48 = 0.

Consider the equation x^{2} + 14 x + 39 = 0.

To find the x-intercepts of a particular parabola, Katrina used the quadratic formula and found that b^{2} - 4 a c = - 5. How many x-intercepts does the parabola have?

For a particular quadratic equation b^{2} - 4 a c = 0. What can be said about the solutions of the quadratic equation?

The solutions of a quadratic equation are 9 and - 9.

What can be said about the value of b^{2} - 4 a c?

Consider the following quadratic equations in terms of x. Find an expression for the discriminant of the equation:

Find the values of n for which x^{2} - 8 n x + 1296 = 0 has one solution.

Consider the equation in terms of x:

Consider the equation in terms of x:

Given that it has two solutions, determine the possible values of m.

The graph of y = m - 9 x - 3 x^{2} has no x-intercepts.

Solve for the value(s) of k for which y = 4 x^{2} - 4 x + k - 15 just touches the x-axis.

The equation \left(m + 4\right) x^{2} + 2 m x + 2 = 0 has a single solution. Find the possible values of m.

Find the range of values of k for which y = x^{2} - \left(k - 2\right) x + k - 2 has two x-intercepts.

For the following equations:

Is the following statement true or false?

'Any quadratic equation that can be solved by completing the square can also be solved by the quadratic formula.'

The standard form of a quadratic equation is a x^{2} + b x + c = 0. Find the values of a, b and c in the quadratic equations below:

Solve the following equations using the quadratic formula:

Solve the following equations using the quadratic formula. Leave your answers in surd form.

Solve the following equations using the quadratic formula. Round your answers to 1 decimal place.

Consider the equation x \left(x + 9\right) = - 20.

Using the quadratic formula, the solutions to a quadratic equation of the form

ax^2 + bx + c = 0 are given by: x = \dfrac{- 5 \pm \sqrt{5^{2} - 4 \times \left( - 7 \right) \times 10}}{2 \times \left( - 7 \right)}

An object is launched from a height of 90 feet with an initial velocity of 131 feet per second. After x seconds, its height (in feet) is given by h = - 16 x^{2} + 131 x + 90

Solve for the number of seconds, x, after which the object is 20 feet above the ground. Give your answer to the nearest tenth of a second.

Consider a right-angled triangle with side lengths x units, x + p units and x + q units, ordered from shortest to longest. No two sides of this triangle have the same length.

a

Complete the statement:

p and q have lengths such that

0<⬚<⬚.

b

Write a quadratic equation that describes the relationship between the sides of the triangle in terms of x.

c

Find the discriminant of this quadratic equation.

d

How many real solutions are there to the quadratic equation?

e

Find the value of x when q = 2 p. Give your answer in terms of p.