Answer :
To solve the quadratic equation \(4w^2 - 19w = 5\), let us proceed step-by-step.
1. Rearrange the equation in standard form:
First, rewrite the equation so that it is in the form \( ax^2 + bx + c = 0 \).
Given:
[tex]\[ 4w^2 - 19w = 5 \][/tex]
Subtract 5 from both sides to set the equation to 0:
[tex]\[ 4w^2 - 19w - 5 = 0 \][/tex]
2. Identify coefficients:
In the equation \(4w^2 - 19w - 5 = 0\),
- \( a = 4 \)
- \( b = -19 \)
- \( c = -5 \)
3. Calculate the discriminant:
The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values of \( a \), \( b \), and \( c \):
[tex]\[ \Delta = (-19)^2 - 4(4)(-5) \][/tex]
[tex]\[ \Delta = 361 + 80 \][/tex]
[tex]\[ \Delta = 441 \][/tex]
4. Determine the number of solutions:
- Since \( \Delta > 0 \), there are two distinct real solutions.
5. Find the solutions using the quadratic formula:
The quadratic formula to find the roots of \( ax^2 + bx + c = 0 \) is:
[tex]\[ w = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Using the discriminant \( \Delta = 441 \):
- The positive root:
[tex]\[ w_1 = \frac{-(-19) + \sqrt{441}}{2 \cdot 4} \][/tex]
[tex]\[ w_1 = \frac{19 + 21}{8} \][/tex]
[tex]\[ w_1 = \frac{40}{8} \][/tex]
[tex]\[ w_1 = 5 \][/tex]
- The negative root:
[tex]\[ w_2 = \frac{-(-19) - \sqrt{441}}{2 \cdot 4} \][/tex]
[tex]\[ w_2 = \frac{19 - 21}{8} \][/tex]
[tex]\[ w_2 = \frac{-2}{8} \][/tex]
[tex]\[ w_2 = -0.25 \][/tex]
6. Write the solutions:
The solutions to the equation \(4w^2 - 19w - 5 = 0\) are:
[tex]\[ w = 5, -0.25 \][/tex]
Therefore, the answers are [tex]\( w = 5 \)[/tex] and [tex]\( w = -0.25 \)[/tex].
1. Rearrange the equation in standard form:
First, rewrite the equation so that it is in the form \( ax^2 + bx + c = 0 \).
Given:
[tex]\[ 4w^2 - 19w = 5 \][/tex]
Subtract 5 from both sides to set the equation to 0:
[tex]\[ 4w^2 - 19w - 5 = 0 \][/tex]
2. Identify coefficients:
In the equation \(4w^2 - 19w - 5 = 0\),
- \( a = 4 \)
- \( b = -19 \)
- \( c = -5 \)
3. Calculate the discriminant:
The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values of \( a \), \( b \), and \( c \):
[tex]\[ \Delta = (-19)^2 - 4(4)(-5) \][/tex]
[tex]\[ \Delta = 361 + 80 \][/tex]
[tex]\[ \Delta = 441 \][/tex]
4. Determine the number of solutions:
- Since \( \Delta > 0 \), there are two distinct real solutions.
5. Find the solutions using the quadratic formula:
The quadratic formula to find the roots of \( ax^2 + bx + c = 0 \) is:
[tex]\[ w = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Using the discriminant \( \Delta = 441 \):
- The positive root:
[tex]\[ w_1 = \frac{-(-19) + \sqrt{441}}{2 \cdot 4} \][/tex]
[tex]\[ w_1 = \frac{19 + 21}{8} \][/tex]
[tex]\[ w_1 = \frac{40}{8} \][/tex]
[tex]\[ w_1 = 5 \][/tex]
- The negative root:
[tex]\[ w_2 = \frac{-(-19) - \sqrt{441}}{2 \cdot 4} \][/tex]
[tex]\[ w_2 = \frac{19 - 21}{8} \][/tex]
[tex]\[ w_2 = \frac{-2}{8} \][/tex]
[tex]\[ w_2 = -0.25 \][/tex]
6. Write the solutions:
The solutions to the equation \(4w^2 - 19w - 5 = 0\) are:
[tex]\[ w = 5, -0.25 \][/tex]
Therefore, the answers are [tex]\( w = 5 \)[/tex] and [tex]\( w = -0.25 \)[/tex].
