Certainly! To find the x-intercepts and y-intercepts for the parabola given by the equation [tex]\( y = 5x^2 - 8x - 4 \)[/tex], we need to follow these steps:
### Finding the y-intercept
To find the y-intercept of a parabola, we set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex].
Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = 5(0)^2 - 8(0) - 4 \][/tex]
[tex]\[ y = -4 \][/tex]
So, the y-intercept is [tex]\( (0, -4) \)[/tex].
### Finding the x-intercepts
To find the x-intercepts, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex].
Set [tex]\( y \)[/tex] to 0 in the equation:
[tex]\[ 0 = 5x^2 - 8x - 4 \][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. We can solve it using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 5 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -4 \)[/tex].
Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = (-8)^2 - 4(5)(-4) = 64 + 80 = 144 \][/tex]
Now, apply the quadratic formula:
[tex]\[ x = \frac{-(-8) \pm \sqrt{144}}{2(5)} \][/tex]
[tex]\[ x = \frac{8 \pm 12}{10} \][/tex]
So we have two solutions:
[tex]\[ x = \frac{8 + 12}{10} = \frac{20}{10} = 2 \][/tex]
[tex]\[ x = \frac{8 - 12}{10} = \frac{-4}{10} = -\frac{2}{5} \][/tex]
So, the x-intercepts are [tex]\( \left(2, 0\right) \)[/tex] and [tex]\( \left(-\frac{2}{5}, 0\right) \)[/tex].
### Summary
- The y-intercept is [tex]\( (0, -4) \)[/tex].
- The x-intercepts are [tex]\( \left(2, 0\right) \)[/tex] and [tex]\( \left(-\frac{2}{5}, 0\right) \)[/tex].
