Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Certainly! Let's prove that the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] has roots given by [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex].
To do this, we will derive the quadratic formula step-by-step.
### Step 1: Write the General Form
Consider the quadratic equation:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
### Step 2: Normalize the Equation
If [tex]\( a \neq 0 \)[/tex], we can divide the entire equation by [tex]\( a \)[/tex] to simplify:
[tex]\[ x^2 + \frac{b}{a} x + \frac{c}{a} = 0 \][/tex]
### Step 3: Complete the Square
To complete the square, we need to make the left side of the equation a perfect square trinomial. Let's isolate the constant term:
[tex]\[ x^2 + \frac{b}{a} x = -\frac{c}{a} \][/tex]
Next, add and subtract [tex]\((\frac{b}{2a})^2\)[/tex] on the left-hand side:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} \][/tex]
This simplifies to:
[tex]\[ \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} \][/tex]
### Step 4: Simplify the Equation
Combine the terms involving [tex]\( x \)[/tex]:
[tex]\[ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{c}{a} \][/tex]
Now, get a common denominator on the right-hand side and simplify:
[tex]\[ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Take the square root of both sides:
[tex]\[ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \][/tex]
Isolate [tex]\( x \)[/tex] by subtracting [tex]\(\frac{b}{2a}\)[/tex] from both sides:
[tex]\[ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 6: Combine the Terms
Combine the terms on the right-hand side under a common denominator:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Thus, we have derived that the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Conclusion:
If [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = 2 \)[/tex], then substituting these values into the quadratic formula gives:
[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(2)}}{2(1)} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{9 - 8}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm 1}{2} \][/tex]
Thus, the roots are:
[tex]\[ x_1 = \frac{3 + 1}{2} = 2 \][/tex]
[tex]\[ x_2 = \frac{3 - 1}{2} = 1 \][/tex]
The discriminant [tex]\( \Delta \)[/tex] here is [tex]\( b^2 - 4ac = 1 \)[/tex], confirming that the roots indeed are [tex]\( 2.0 \)[/tex] and [tex]\( 1.0 \)[/tex].
To do this, we will derive the quadratic formula step-by-step.
### Step 1: Write the General Form
Consider the quadratic equation:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
### Step 2: Normalize the Equation
If [tex]\( a \neq 0 \)[/tex], we can divide the entire equation by [tex]\( a \)[/tex] to simplify:
[tex]\[ x^2 + \frac{b}{a} x + \frac{c}{a} = 0 \][/tex]
### Step 3: Complete the Square
To complete the square, we need to make the left side of the equation a perfect square trinomial. Let's isolate the constant term:
[tex]\[ x^2 + \frac{b}{a} x = -\frac{c}{a} \][/tex]
Next, add and subtract [tex]\((\frac{b}{2a})^2\)[/tex] on the left-hand side:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} \][/tex]
This simplifies to:
[tex]\[ \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} \][/tex]
### Step 4: Simplify the Equation
Combine the terms involving [tex]\( x \)[/tex]:
[tex]\[ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{c}{a} \][/tex]
Now, get a common denominator on the right-hand side and simplify:
[tex]\[ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Take the square root of both sides:
[tex]\[ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \][/tex]
Isolate [tex]\( x \)[/tex] by subtracting [tex]\(\frac{b}{2a}\)[/tex] from both sides:
[tex]\[ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 6: Combine the Terms
Combine the terms on the right-hand side under a common denominator:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Thus, we have derived that the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Conclusion:
If [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = 2 \)[/tex], then substituting these values into the quadratic formula gives:
[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(2)}}{2(1)} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{9 - 8}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm 1}{2} \][/tex]
Thus, the roots are:
[tex]\[ x_1 = \frac{3 + 1}{2} = 2 \][/tex]
[tex]\[ x_2 = \frac{3 - 1}{2} = 1 \][/tex]
The discriminant [tex]\( \Delta \)[/tex] here is [tex]\( b^2 - 4ac = 1 \)[/tex], confirming that the roots indeed are [tex]\( 2.0 \)[/tex] and [tex]\( 1.0 \)[/tex].
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
