To solve the quadratic equation [tex]\(x^2 + 2x - 1 = 2\)[/tex], we first need to rewrite it in standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
Step 1: Rewrite the equation in standard form
We start with:
[tex]\[ x^2 + 2x - 1 = 2 \][/tex]
First, subtract 2 from both sides to set the equation to zero:
[tex]\[ x^2 + 2x - 1 - 2 = 0 \][/tex]
[tex]\[ x^2 + 2x - 3 = 0 \][/tex]
Now, the equation is in standard form [tex]\(ax^2 + bx + c = 0\)[/tex], with [tex]\(a = 1\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -3\)[/tex].
Step 2: Identify the coefficients
For the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 2 \][/tex]
[tex]\[ c = -3 \][/tex]
Step 3: Calculate the discriminant
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 2^2 - 4(1)(-3) \][/tex]
[tex]\[ \Delta = 4 + 12 \][/tex]
[tex]\[ \Delta = 16 \][/tex]
Step 4: Apply the quadratic formula
The solutions of the quadratic equation are given by the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[ x = \frac{-(2) \pm \sqrt{16}}{2(1)} \][/tex]
[tex]\[ x = \frac{-2 \pm 4}{2} \][/tex]
Step 5: Calculate the two possible solutions
1. For the positive square root:
[tex]\[ x = \frac{-2 + 4}{2} \][/tex]
[tex]\[ x = \frac{2}{2} \][/tex]
[tex]\[ x = 1 \][/tex]
2. For the negative square root:
[tex]\[ x = \frac{-2 - 4}{2} \][/tex]
[tex]\[ x = \frac{-6}{2} \][/tex]
[tex]\[ x = -3 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\(x^2 + 2x - 3 = 0\)[/tex] are:
[tex]\[ x = 1 \][/tex]
and
[tex]\[ x = -3 \][/tex]
In conclusion, the roots of the quadratic equation [tex]\(x^2 + 2x - 3 = 0\)[/tex] are [tex]\(x = 1\)[/tex] and [tex]\(x = -3\)[/tex]. The discriminant in this case is [tex]\(16\)[/tex], confirming that there are two distinct real solutions.
