Question
Assignment for what value of \( R \) does the quadratic equation \( 3 x^{2}- \) \( 7 x+R \) has equal rooss.
Ask by French Davey. in Nigeria
Feb 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The value of \( R \) is \( \frac{49}{12} \) or approximately \( 4.083 \).
Solution
To find the value of \( R \) for which the quadratic equation \( 3x^{2} - 7x + R \) has equal roots, we need to set the discriminant of the quadratic equation to zero.
The discriminant of a quadratic equation \( ax^{2} + bx + c \) is given by the formula \( b^{2} - 4ac \).
For the given quadratic equation \( 3x^{2} - 7x + R \), the coefficients are \( a = 3 \), \( b = -7 \), and \( c = R \).
Setting the discriminant to zero, we have:
\[ (-7)^{2} - 4(3)(R) = 0 \]
Simplifying the equation, we get:
\[ 49 - 12R = 0 \]
Now, we can solve for \( R \) by isolating it on one side of the equation.
Solve the equation by following steps:
- step0: Solve for \(R\):
\(49-12R=0\)
- step1: Move the constant to the right side:
\(-12R=0-49\)
- step2: Remove 0:
\(-12R=-49\)
- step3: Change the signs:
\(12R=49\)
- step4: Divide both sides:
\(\frac{12R}{12}=\frac{49}{12}\)
- step5: Divide the numbers:
\(R=\frac{49}{12}\)
The value of \( R \) for which the quadratic equation \( 3x^{2} - 7x + R \) has equal roots is \( R = \frac{49}{12} \) or approximately \( R = 4.083 \).
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Simplify this solution Mind Expander
For a quadratic equation in the form \( ax^2 + bx + c = 0 \) to have equal roots, the discriminant must be zero. The discriminant \( D \) is given by \( D = b^2 - 4ac \). In your equation \( 3x^2 - 7x + R = 0 \), we have \( a = 3 \), \( b = -7 \), and \( c = R \). Setting the discriminant to zero gives us: \[ D = (-7)^2 - 4(3)(R) = 49 - 12R = 0 \] Now, solve for \( R \): \[ 49 = 12R \implies R = \frac{49}{12} \] Thus, the value of \( R \) for which the quadratic equation has equal roots is \( \frac{49}{12} \).
