1) 2x^2 + 7x + 3 = 0; 2) x^2 - 8x + 16 = 0; 3) 2x^2 - 4x + 5 = 0.
Understand the Problem
The question contains multiple quadratic equations along with their solutions. It demonstrates how to calculate the discriminant, roots, and other necessary steps to solve these equations. This is mainly focused on solving quadratic equations using the quadratic formula.
Answer
1) Roots: $x_1 = -3$, $x_2 = -\frac{1}{2}$; 2) Root: $x = 4$; 3) No real roots.
Answer for screen readers
-
Roots: $x_1 = -3$, $x_2 = -\frac{1}{2}$
-
Root: $x = 4$
-
No real roots.
Steps to Solve
-
Identify coefficients in the quadratic equation For each equation of the form $ax^2 + bx + c = 0$, determine the coefficients:
- For the first equation, $a = 2$, $b = 7$, $c = 3$.
-
Calculate the discriminant (D) Use the formula for the discriminant: $$ D = b^2 - 4ac $$
- For the first equation: $$ D = 7^2 - 4 \cdot 2 \cdot 3 = 49 - 24 = 25 $$
-
Find the roots using the quadratic formula The roots can be calculated using $$ x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} $$
- For $x_1$ and $x_2$:
- $x_1 = \frac{-7 - 5}{2 \cdot 2} = \frac{-12}{4} = -3$
- $x_2 = \frac{-7 + 5}{2 \cdot 2} = \frac{-2}{4} = -\frac{1}{2}$
-
Repeat the process for the second equation For the equation $x^2 - 8x + 16 = 0$:
- Here $a = 1$, $b = -8$, $c = 16$.
- Calculate the discriminant: $$ D = (-8)^2 - 4 \cdot 1 \cdot 16 = 64 - 64 = 0 $$
- Since $D = 0$, there is one real root: $$ x = \frac{-b}{2a} = \frac{8}{2} = 4 $$
-
Evaluate the third equation For the equation $2x^2 - 4x + 5 = 0$:
- Here $a = 2$, $b = -4$, $c = 5$.
- Calculate the discriminant: $$ D = (-4)^2 - 4 \cdot 2 \cdot 5 = 16 - 40 = -24 $$
- Since $D < 0$, there are no real roots.
-
Roots: $x_1 = -3$, $x_2 = -\frac{1}{2}$
-
Root: $x = 4$
-
No real roots.
More Information
The quadratic formula can yield different types of solutions based on the discriminant: two real roots if $D > 0$, one real root if $D = 0$, and no real roots if $D < 0$. The process of solving quadratic equations is fundamental in algebra, with applications in various fields.
Tips
- Confusing the signs in the quadratic formula can lead to incorrect roots.
- Failing to simplify the discriminant correctly can result in an incorrect understanding of the number of roots.
AI-generated content may contain errors. Please verify critical information
