Firstly, starting with the y-intercept. To find the y-intercept, set the x variable to zero and solve as such:
[tex]y=3*0^2+24*0-51\\y=0+0-51\\y=-51[/tex]
Your y-intercept is (0,-51).
Next, using our equation plug the appropriate values into the quadratic formula:
[tex]x=\frac{-24\pm \sqrt{24^2-4*3*(-54)}}{2*3}[/tex]
Next, solve the multiplications and exponent:
[tex]x=\frac{-24\pm \sqrt{576-(-648)}}{6}\\\\x=\frac{-24\pm \sqrt{576+648}}{6}[/tex]
Next, solve the addition:
[tex]x=\frac{-24\pm \sqrt{1224}}{6}[/tex]
Now, simplify the radical using the product rule of radicals as such:
√1224 = √12 × √102 = √2 × √6 × √6 × √17 = 6 × √2 × √17 = 6√34
[tex]x=\frac{-24\pm 6\sqrt{34}}{6}[/tex]
Next, divide:
[tex]x=-4\pm \sqrt{34}[/tex]
The exact values of your x-intercepts are (-4 + √34, 0) and (-4 - √34, 0).
Now to find the approximate values, solve this twice: once with the + symbol and once with the - symbol:
[tex]x=-4+ \sqrt{34},-4- \sqrt{34}\\x\approx 1.83, -9.83[/tex]
The approximate values of your x-intercepts (rounded to the hundredths) are (1.83,0) and (-9.83,0).
