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Sagot :
Certainly! Let's walk through the process step-by-step in detail.
We start with the general quadratic equation:
1. Statement: [tex]\( ax^2 + bx + c = 0 \)[/tex]
Reason: Given
2. Statement: [tex]\( ax^2 + bx = -c \)[/tex]
Reason: Subtract [tex]\( c \)[/tex] from both sides of the equation
3. Statement: [tex]\( x^2 + \frac{b}{a}x = -\frac{c}{a} \)[/tex]
Reason: Divide both sides of the equation by [tex]\( a \)[/tex]
4. Statement: [tex]\( x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \)[/tex]
Reason: Complete the square by adding [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] to both sides
5. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} \)[/tex]
Reason: Square [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] on the right side of the equation
6. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} \)[/tex]
Reason: Find a common denominator on the right side of the equation
7. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \)[/tex]
Reason: Add the fractions together on the right side of the equation
8. Statement: [tex]\( \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \)[/tex]
Reason: Rewrite the left side as a square of the binomial
By following these steps, we transform the quadratic equation into a form that allows us to recognize the left side as a perfect square trinomial, making it easier to further solve the equation using the quadratic formula.
We start with the general quadratic equation:
1. Statement: [tex]\( ax^2 + bx + c = 0 \)[/tex]
Reason: Given
2. Statement: [tex]\( ax^2 + bx = -c \)[/tex]
Reason: Subtract [tex]\( c \)[/tex] from both sides of the equation
3. Statement: [tex]\( x^2 + \frac{b}{a}x = -\frac{c}{a} \)[/tex]
Reason: Divide both sides of the equation by [tex]\( a \)[/tex]
4. Statement: [tex]\( x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \)[/tex]
Reason: Complete the square by adding [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] to both sides
5. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} \)[/tex]
Reason: Square [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] on the right side of the equation
6. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} \)[/tex]
Reason: Find a common denominator on the right side of the equation
7. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \)[/tex]
Reason: Add the fractions together on the right side of the equation
8. Statement: [tex]\( \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \)[/tex]
Reason: Rewrite the left side as a square of the binomial
By following these steps, we transform the quadratic equation into a form that allows us to recognize the left side as a perfect square trinomial, making it easier to further solve the equation using the quadratic formula.
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