Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Certainly! Let's solve the quadratic equation [tex]\( x^2 + 3x - 54 = 0 \)[/tex] step-by-step.
### Step 1: Write down the quadratic equation
[tex]\[ x^2 + 3x - 54 = 0 \][/tex]
### Step 2: Identify the coefficients
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. From the given equation, we identify the coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( c = -54 \)[/tex]
### Step 3: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] 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]\[ \Delta = 3^2 - 4(1)(-54) \][/tex]
[tex]\[ \Delta = 9 + 216 \][/tex]
[tex]\[ \Delta = 225 \][/tex]
### Step 4: Determine the roots using the quadratic formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] into the formula:
[tex]\[ x = \frac{-3 \pm \sqrt{225}}{2(1)} \][/tex]
[tex]\[ x = \frac{-3 \pm 15}{2} \][/tex]
### Step 5: Calculate the two possible solutions
1. For the positive square root:
[tex]\[ x = \frac{-3 + 15}{2} = \frac{12}{2} = 6 \][/tex]
2. For the negative square root:
[tex]\[ x = \frac{-3 - 15}{2} = \frac{-18}{2} = -9 \][/tex]
### Step 6: Write down the solutions
The solutions to the equation [tex]\( x^2 + 3x - 54 = 0 \)[/tex] are:
[tex]\[ x = -9 \quad \text{and} \quad x = 6 \][/tex]
So, the solutions are [tex]\( x = -9 \)[/tex] and [tex]\( x = 6 \)[/tex].
### Step 1: Write down the quadratic equation
[tex]\[ x^2 + 3x - 54 = 0 \][/tex]
### Step 2: Identify the coefficients
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. From the given equation, we identify the coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( c = -54 \)[/tex]
### Step 3: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] 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]\[ \Delta = 3^2 - 4(1)(-54) \][/tex]
[tex]\[ \Delta = 9 + 216 \][/tex]
[tex]\[ \Delta = 225 \][/tex]
### Step 4: Determine the roots using the quadratic formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] into the formula:
[tex]\[ x = \frac{-3 \pm \sqrt{225}}{2(1)} \][/tex]
[tex]\[ x = \frac{-3 \pm 15}{2} \][/tex]
### Step 5: Calculate the two possible solutions
1. For the positive square root:
[tex]\[ x = \frac{-3 + 15}{2} = \frac{12}{2} = 6 \][/tex]
2. For the negative square root:
[tex]\[ x = \frac{-3 - 15}{2} = \frac{-18}{2} = -9 \][/tex]
### Step 6: Write down the solutions
The solutions to the equation [tex]\( x^2 + 3x - 54 = 0 \)[/tex] are:
[tex]\[ x = -9 \quad \text{and} \quad x = 6 \][/tex]
So, the solutions are [tex]\( x = -9 \)[/tex] and [tex]\( x = 6 \)[/tex].
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.