For what values of k, the roots of 9x 2 + 8kx + 16 = 0 are real and equal?
-3,3
Finding k for Real and Equal Roots
To find the values of k for which the roots of a quadratic equation are real and equal, we need to use the concept of the discriminant. A quadratic equation in the standard form is given by \(ax^2 + bx + c = 0\). The nature of the roots depends on the value of the discriminant, denoted by \(\Delta\) or D, which is calculated as \(b^2 - 4ac\).
For a quadratic equation to have real and equal roots, the discriminant must be exactly equal to zero (\(\Delta = 0\)).
Identifying Coefficients
The given quadratic equation is \(9x^2 + 8kx + 16 = 0\). Comparing this with the standard form \(ax^2 + bx + c = 0\), we can identify the coefficients:
- a = 9 (coefficient of \(x^2\))
- b = 8k (coefficient of x)
- c = 16 (constant term)
Setting the Discriminant to Zero
We know that for real and equal roots, the discriminant \(\Delta\) must be zero.
So, we set the discriminant formula equal to 0: \[ \Delta = b^2 - 4ac = 0 \]
Solving for k
Now, substitute the values of a, b, and c into the equation \(\Delta = 0\): \[ (8k)^2 - 4(9)(16) = 0 \]
Let's simplify and solve for k: \[ 64k^2 - 4 \times 9 \times 16 = 0 \] \[ 64k^2 - 36 \times 16 = 0 \] Calculate \(36 \times 16\): \[ 36 \times 16 = 576 \] Substitute this value back into the equation: \[ 64k^2 - 576 = 0 \] Now, isolate the \(k^2\) term: \[ 64k^2 = 576 \] Divide both sides by 64: \[ k^2 = \frac{576}{64} \] Perform the division: \[ \frac{576}{64} = 9 \] So, we have: \[ k^2 = 9 \] To find the values of k, take the square root of both sides: \[ k = \pm \sqrt{9} \] \[ k = \pm 3 \] This means there are two possible values for k that make the roots of the equation real and equal: \(k = 3\) and \(k = -3\).
Conclusion on the Value of k
The values of k for which the roots of the equation \(9x^2 + 8kx + 16 = 0\) are real and equal are -3 and 3.
| Equation | \(ax^2 + bx + c = 0\) |
|---|---|
| Given Equation | \(9x^2 + 8kx + 16 = 0\) |
| Coefficient a | 9 |
| Coefficient b | 8k |
| Coefficient c | 16 |
| Condition for Real & Equal Roots | Discriminant \(\Delta = b^2 - 4ac = 0\) |
| Discriminant Calculation | \((8k)^2 - 4(9)(16) = 0\) |
| Resulting Equation for k | \(64k^2 - 576 = 0\) |
| Values of k | \(k = \pm 3\) |
Revision Table: Quadratic Roots and Discriminant
| Discriminant (\(\Delta = b^2 - 4ac\)) | Nature of Roots |
|---|---|
| \(\Delta > 0\) | Two distinct real roots |
| \(\Delta = 0\) | Two real and equal roots |
| \(\Delta < 0\) | Two complex (non-real) roots |
Additional Information on Quadratic Equations
A quadratic equation is a polynomial equation of the second degree. The roots of a quadratic equation are the values of the variable (usually x) that satisfy the equation. These roots represent the x-intercepts of the parabola represented by the quadratic function \(y = ax^2 + bx + c\).
The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), can also be used to find the roots directly. The expression under the square root in this formula is the discriminant.
- If the discriminant is positive, the square root is a positive real number, leading to two different real roots (\(\frac{-b + \sqrt{\Delta}}{2a}\) and \(\frac{-b - \sqrt{\Delta}}{2a}\)).
- If the discriminant is zero, the square root is zero, leading to only one value for the root, which is \(\frac{-b}{2a}\). Since the quadratic equation has degree 2, this root is counted twice, hence "two real and equal roots".
- If the discriminant is negative, the square root is an imaginary number, leading to two complex conjugate roots.
Understanding the discriminant is a fundamental concept in algebra for analyzing the nature of the roots of quadratic equations without actually solving for them.
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