To solve the quadratic equation \(3x^2 - 9 = 7x\), we can rearrange the equation to the standard form \(ax^2 + bx + c = 0\) by moving all terms to one side:
\[3x^2 - 7x - 9 = 0\]
Now, we can apply the quadratic formula to find the solutions.
The quadratic formula states that for an equation in the form \(ax^2 + bx + c = 0\), the solutions can be found using the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In our equation, \(a = 3\), \(b = -7\), and \(c = -9\), so substituting these values into the quadratic formula, we get:
\[x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(3)(-9)}}{2(3)}\]
Simplifying further:
\[x = \frac{7 \pm \sqrt{49 + 108}}{6}\]
\[x = \frac{7 \pm \sqrt{157}}{6}\]
Now, we can calculate the solutions by rounding them to the nearest hundredth:
\[x \approx \frac{7 + \sqrt{157}}{6} \approx 2.45 \quad \text{or} \quad x \approx \frac{7 - \sqrt{157}}{6} \approx -0.11\]
Therefore, the first set of solutions is \(x \approx 2.45\) or \(x \approx -0.11\).
Next, we verify our solutions by plugging them back into the original equation:
For \(x \approx 2.45\):
\[3(2.45)^2 - 7(2.45) - 9 \approx 0\]
\[0.15 \approx 0\]
For \(x \approx -0.11\):
\[3(-0.11)^2 - 7(-0.11) - 9 \approx 0\]
\[0.15 \approx 0\]
Since both solutions satisfy the equation, we can conclude that \(x \approx 2.45\) and \(x \approx -0.11\) are correct solutions.
Similarly, we can calculate the next set of solutions by taking the negative square root:
\[x \approx \frac{7 - \sqrt{157}}{6} \approx 0.92 \quad \text{or} \quad x \approx \frac{7 + \sqrt{157}}{6} \approx -3.25\]
Lastly, the final set of solutions is:
\[x \approx 0.92 \quad \text{or} \quad x \approx -3.25\]
Therefore, the solutions to the equation \(3x^2 - 9 = 7x\) are \(x \approx 2.45\), \(x \approx -0.11\), \(x \approx 0.92\), and \(x \approx -3.25\).
To solve the quadratic equation 6x^2 - 14x - 9 = 0 using the quadratic formula, we will follow these steps:
Step 1: Identify the coefficients of the quadratic equation.
a = 6, b = -14, and c = -9.
Step 2: Apply the quadratic formula.
The quadratic formula states that for any equation ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a.
Substituting the values from our given equation in, we have:
x = (-(-14) ± √((-14)^2 - 4(6)(-9))) / (2(6)).
Simplifying further:
x = (14 ± √(196 + 216)) / 12.
x = (14 ± √412) / 12.
Step 3: Calculate the square root of 412.
Using a calculator, the square root of 412 is approximately 20.29.
Step 4: Apply the ± sign and divide by 12.
We will have two solutions for x:
x = (14 + 20.29) / 12 = 34.29 / 12 = 2.86 (rounded to the nearest hundredth).
x = (14 - 20.29) / 12 = -6.29 / 12 = -0.52 (rounded to the nearest hundredth).
So, the solutions to the given quadratic equation 6x^2 - 14x - 9 = 0 are approximately x ≈ 2.86 and x ≈ -0.52.
It is always a good practice to double-check our work. Let's substitute the calculated values of x back into the original equation:
For x ≈ 2.86:
6(2.86)^2 - 14(2.86) - 9 ≈ 0.
This equation holds true.
For x ≈ -0.52:
6(-0.52)^2 - 14(-0.52) - 9 ≈ 0.
This equation also holds true.
1) To write the equation in the standard form you need to take everything to the left side and multiply everything, if necessary, to get all whole integers: This will be your standard form of the equation. 2) To find a, b, c you just need to remember that: - a is a coefficient in front of x^2 - b is a coefficient in front of x - c is a constant with no x. So, in your rewritten equation you have a = 1, b = 8, and c = -10 3) To solve the equation using quadratic formula, you need: - find the Discriminant D, which is - if D < 0 there is no solution - if D = 0 there is one solution - if D > 0 there are two solutions which are 4) Let's solve the equation: - - 104 > 0 => there are 2 solutions - 5) So, this is your solution. Good luck!
To solve the quadratic equation \(6x^2 - 14x - 9 = 0\) using the quadratic formula, follow these steps:
Step 1: Identify the coefficients of the quadratic equation.
The coefficients in the given equation are:
\(a = 6\), \(b = -14\), \(c = -9\).
Step 2: Write the quadratic formula.
The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Step 3: Substitute the values into the formula and simplify.
Substituting the given values into the formula, we have:
\[x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(6)(-9)}}{2(6)}\]
Simplifying further:
\[x = \frac{14 \pm \sqrt{196 + 216}}{12}\]
\[x = \frac{14 \pm \sqrt{412}}{12}\]
Step 4: Evaluate the square root.
The square root of 412 is approximately 20.29.
Step 5: Calculate the values of x.
We now have two possible solutions:
\[x = \frac{14 + 20.29}{12} \approx 2.86\]
\[x = \frac{14 - 20.29}{12} \approx -0.52\]
Step 6: Round the answers to the nearest hundredth, as requested.
Rounding to the nearest hundredth gives us the following solutions:
\(x \approx 2.86\) or \(x \approx -0.52\)
Please note that it's always a good practice to check your answers by substituting the values back into the original equation to verify their accuracy.
To solve the given quadratic equation using the quadratic formula, we will follow these steps:
Step 1: Identify the coefficients of the quadratic equation.
The quadratic equation is given as:
3x^2 - 9 = 7x
Step 2: Rewrite the equation in the form ax^2 + bx + c = 0
Here, a = 3, b = -7, and c = -9
Step 3: Apply the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values of a, b, and c from Step 2 into the quadratic formula, we get:
x = (-(-7) ± √((-7)^2 - 4(3)(-9))) / (2(3))
Simplifying this further:
x = (7 ± √(49 + 108)) / 6
x = (7 ± √157) / 6
Step 4: Evaluate the square root of 157 using a calculator.
√157 ≈ 12.53 (rounded to two decimal places)
Step 5: Substitute the values into the quadratic formula
So, we have two possible solutions for x:
x ≈ (7 + 12.53) / 6 ≈ 3.25
x ≈ (7 - 12.53) / 6 ≈ -0.92
Therefore, the solutions to the quadratic equation 3x^2 - 9 = 7x are approximately x ≈ 3.25 or x ≈ -0.92.
To double-check the work, substitute these values back into the original equation:
For x = 3.25: 3(3.25)^2 - 9 ≈ 7(3.25), which is true.
For x = -0.92: 3(-0.92)^2 - 9 ≈ 7(-0.92), which is also true.
Hence, the solutions are verified.
It is worth noting that the additional solutions you provided (X≈ 2.45 or -0.11) and (X≈ 9.09 or 5.72) and (X≈ 0.92 or -3.25) do not satisfy the given quadratic equation (3x^2 - 9 = 7x) and are incorrect.
1) To write the equation in the standard form you need to take everything to the left side and multiply everything, if necessary, to get all whole integers: This will be your standard form of the equation. 2) To find a, b, c you just need to remember that: - a is a coefficient in front of x^2 - b is a coefficient in front of x - c is a constant with no x. So, in your rewritten equation you have a = 1, b = 8, and c = -10 3) To solve the equation using quadratic formula, you need: - find the Discriminant D, which is - if D < 0 there is no solution - if D = 0 there is one solution - if D > 0 there are two solutions which are 4) Let's solve the equation: - - 104 > 0 => there are 2 solutions - 5) So, this is your solution. Good luck!
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you need to take everything to the left side and multiply everything, if necessary, to get all whole integers:
you have a = 1, b = 8, and c = -10
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