fill in the blanks. solve 2x^2+3x-5=0

Question

ax^2+bx+c=0
a=___ b=___ c=___
x= -___± √__^2 - 4___ ___/2___
simplify -b= ___ b^2- 4ac = ___ 2a = ___
replace x= __ ± √___/___
x= ___ ___

Answer 1

To solve the quadratic equation $2x^2+3x-5=0$, we can fill in the blanks as follows:

The standard form of a quadratic equation is: $$a{x}^2+bx+c=0$$ For the equation $2x^2+3x-5=0$:

• $a=2$
• $b=3$
• $c=-5$

Now, we apply the quadratic formula: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Filling in the blanks:

• $-b=-3$
• $b^2-4ac=3^2-4(2)(-5)=9+40=49$
• $2a=2(2)=4$

Thus, we can express the previous formula in filled blanks: $$x=\frac{-3\pm \sqrt{49}}{4}$$

Now we simplify:

• $\sqrt{49}=7$

So we have: $$x=\frac{-3\pm 7}{4}$$

Now we can calculate the two possible solutions:

1. $x=\frac{-3+7}{4}=\frac{4}{4}=1$
2. $x=\frac{-3-7}{4}=\frac{-10}{4}=-\frac{5}{2}$

Thus, the final solutions are: $$x=1\text{and}x=-\frac{5}{2}$$

Summarizing the entire solution, we have:

• $a=2$
• $b=3$
• $c=-5$
• $-b=-3$
• $b^2-4ac=49$
• $2a=4$
• $x=\frac{-3\pm 7}{4}$
• Solutions: $x=1$ and $x=-\frac{5}{2}$