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?
\left(x - 4\right)^{2} = 0
x^{2} = 9
For the following equations:
Find the value of the discriminant.
x^{2} + 6 x + 9 = 0
4 x^{2} - 6 x + 7 = 0
2 x^{2} - 2 x = x - 1
Consider the equation x^{2} + 22 x + 121 = 0.
Find the value of the discriminant.
State whether the solutions to the equation are rational or irrational.
State whether each of the following equations have any real solutions:
For the following equations:
Find the discriminant.
Real or not real
Rational or irrational
Equal or unequal
5 x^{2} + 4 x + 8 = 0
4 x^{2} + 4 x - 6 = 0
4 x^{2} - 4 x + 1 = 0
Consider the equation x^{2} - 8 x - 48 = 0.
Find the discriminant.
Describe the nature of the roots.
Find the solutions of the equation.
Consider the equation x^{2} + 14 x + 39 = 0.
Find the discriminant.
Describe the nature of the roots.
Find the exact roots of the equation.
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:
m x^{2} + 3 x - 2 = 0
x^{2} + 5 x + p - 5 = 0
Find the values of n for which x^{2} - 8 n x + 1296 = 0 has one solution.
Consider the equation in terms of x:
m x^{2} - 3 x - 5 = 0Given that it has two unique solutions, determine the possible values of m.
There is one value of m that must be eliminated from the range of solutions found in the previous part. What is this value?
Consider the equation in terms of x:
m x^{2} + 2 x - 1 = 0Given 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 possible values of m.
What is the largest possible integer value of m?
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:
Find the discriminant in terms of k.
Find the value of k for when the equation has equal solutions.
Find the value of k for when the equation has real solutions.
Find the value of k for when the equation has no real solutions.
Find the value of k for when the equation has real and distinct solutions.
2 x^{2} + 8 x + k = 0
\left(k + 4\right) x^{2} + 10 x + 3 = 0
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:
x^{2} + 7 x + 10 = 0
4 x^{2} + 3 x = 5
3 x^{2} - 8 x + 2 = 9 x - 7
Solve the following equations using the quadratic formula:
x^{2} + 11 x + 28 = 0
x^{2} - 5 x + 6 = 0
4 x^{2} - 7 x - 15 = 0
x^{2} + 5 x + \dfrac{9}{4} = 0
- 6 - 13 x + 5 x^{2} = 0
Solve the following equations using the quadratic formula. Leave your answers in surd form.
x^{2} - 5 x - 2 = 0
4 x^{2} - x - 10 = 0
- 2 x^{2} - 15 x - 4 = 0
Solve the following equations using the quadratic formula. Round your answers to one decimal place.
1.8 x^{2} + 5.2 x - 2.3 = 0
x^{2} + 7 x - 3 = 0
3 x \left(x + 4\right) = - 3 x + 4
Consider the equation x \left(x + 9\right) = - 20.
Solve it by the method of factorisation.
Check your solution by solving it using the quadratic formula.
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)}
Find the values of a, b and c.
Write down the quadratic equation that has these solutions.
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.
How many real values of x are possible? Explain your answer.
Find the value of x when q = 2 p. Give your answer in terms of p.
