4.09 Discriminant and parabolas (Extended)
For the following graphs of functions of the form y = a x^{2} + b x + c :
State whether the vertex of the parabola is a maximum or minimum point.
State whether the value of a is negative or positive.
State the number of solutions to the equation a x^{2} + b x + c = 0.
Below is the result after using the quadratic formula to solve an equation:
x = \dfrac{- \left( - 10 \right) \pm \sqrt{ - 1 }}{8}
What can be concluded about the solutions of the equation?
By inspection, determine the number of real solutions for each of the following quadratic equations:
For the following quadratic 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
x^{2} - 4 = 0
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.
For the following equations:
Find the value of the discriminant.
5 x^{2} + 4 x + 8 = 0
4 x^{2} + 4 x - 6 = 0
4 x^{2} - 4 x + 1 = 0
x^{2} - 16 = 0
9x^{2} - 42 x = -49
2 x^{2} - 9 x = -2
x^2 = -6x -10
\dfrac{1}{4}x^{2} - 2 x + \dfrac{1}{4} = 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.
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?
Find an expression for the discriminant of the following quadratic equations:
m x^{2} + 3 x - 2 = 0
x^{2} - 4 n x - 2 = 0
x^{2} + 5 x + p - 5 = 0
-px^{2} + 4 x + p = 0
mx^{2} - 2m x + m - 3 = 0
\dfrac{3n}{4}x^{2} + \dfrac{1n}{2} x + 2n = 0
Find the range of values of a for which the quadratic equation a x^{2} - b x + c = 0, where b = 6 and c = 10, has two distinct real solutions.
Consider the equation in terms of x:
m x^{2} + 2 x - 1 = 0Given that it has two unique solutions, determine the possible values of m.
Consider the equation in terms of x:
m x^{2} - 3 x - 5 = 0Given that the equation has two unique solutions, determine the possible values of m.
State the value of m that must be eliminated from the range of solutions found in the previous part. Explain your answer.
Find the values of n for which x^{2} - 8 n x + 1296 = 0 has one solution.
Find the value of k for which 16 x^{2} + 8 x + k = 0 has equal roots.
Find the values of m for which \left(m + 4\right) x^{2} + 2 m x + 2 = 0 has a single solution.
For the following equations:
Find the discriminant in terms of k.
Find the value of k so that the equation has equal solutions.
Find the value of k so that the equation has real solutions.
Find the value of k so that the equation has no real solutions.
Find the value of k so that the equation has real and distinct solutions.
2 x^{2} + 8 x + k = 0
\left(k + 4\right) x^{2} + 10 x + 3 = 0
Consider the equation in terms of x:x^{2} + 18 x + k + 7 = 0
Find the values of k for which the equation has no real solutions.
State the smallest possible integer value of k.
When graphing 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?
When graphing a particular parabola, Tony used the quadratic formula and found that \\ b^{2} - 4 a c = 0. How many x-intercepts does the parabola have?
For each of the following graphs of a quadratic f \left( x \right) = a x^{2} + b x + c, with discriminant \\ \Delta = b^{2} - 4ac :
State whether a \gt 0 or a \lt 0.
State whether \Delta \gt 0, \Delta \lt 0 or \Delta = 0.
Consider the graph of the quadratic function:y = m - 9 x - 3 x^{2}
Find the possible values of m, if the graph has no x-intercepts.
State the largest possible integer value of m.
Determine the value(s) of k for which the graph of y = 4 x^{2} - 4 x + k - 15 just touches the \\ x-axis.
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.
Complete the statement:
p and q have lengths such that
0<⬚<⬚.
Write a quadratic equation that describes the relationship between the sides of the triangle in terms of x.
Find the discriminant of this quadratic equation.
State the number of real solutions to the quadratic equation. Explain your answer.
Find the value of x in terms of p when q = 2 p.
