Step-by-step explanation:
a) The quadratic formula can help find answers to a quadratic equation when it is equal to 0. The formula is [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex] for a quadratic equation of the form [tex]ax^2+bx+c=0[/tex].
In this question, a is 5, b is -6, and c is -2. Let's put them in the equation and solve.
[tex]x=\frac{6\pm\sqrt{(-6)^2-4(5)(-2)}}{2(5)}\\x=\frac{6\pm\sqrt{36+40}}{10}\\x=\frac{6\pm\sqrt{76}}{10}\\x=\frac{6\pm2\sqrt{19}}{10}\\x=\frac{3\pm\sqrt{19}}{5}[/tex]
This means that the value of x is both [tex]\frac{3+\sqrt{19}}5[/tex] and [tex]\frac{3-\sqrt{19}}5[/tex], since both values make x equal to 0. Putting both into the calculator, we get that [tex]x=1.5[/tex] or [tex]x=-0.3[/tex].
b) We can easily solve this equation using factoring, which turns a quadratic into two factors, and finds the solutions by setting both to 0. This will only work if the quadratic is equal to 0.
[tex]x^2+3x-40=0[/tex]
Now, we have to find two numbers that add to 3 and multiply to -40. The numbers 8 and -5 work, as 8-5=3 and 8*-5 is -40.
we can now make our factors x+8 and x-5
[tex](x+8)(x-5)=0[/tex]
If two numbers, say A and B, are being multiplied and equal 0, then either A is 0, or b is 0, or both. Similarly, if x+8 and x-5 are being multiplied and equal 0, either x+8 = 0, or x-5=0.
[tex]x+8=0\\x=-8[/tex] [tex]x-5=0\\x=5[/tex]
This makes our solutions x=-8 and x=5.
If you are not familiar with factoring or the process I have went through above, I highly recommend learning about it.
