Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$. Hence or otherwise, solve the equation $3^{2m+1} = 35 - 8(3^{m})$, where $m \in \mathbb{R}$. Give your answer in the form $m = \log_b p - q$, where $p, q \in \mathbb{N}$.

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Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$ - Leaving Cert Mathematics - Question b - 2021

Question b

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Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$. Hence or otherwise, solve the equation $3^{2m+1} = 35 - 8(3^{m})$, ... show full transcript

Worked Solution & Example Answer:Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$ - Leaving Cert Mathematics - Question b - 2021

Step 1

Find the two roots of $f(x) = 0$

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Answer

To find the roots of the quadratic equation $3x^2 + 8x - 35 = 0$ , we can apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 3$ , $b = 8$ , and $c = -35$ . First, we calculate the discriminant:

$b^2 - 4ac = 8^2 - 4 \cdot 3 \cdot (-35) = 64 + 420 = 484.$

Then we plug this into the quadratic formula:

$x = \frac{-8 \pm \sqrt{484}}{2 \cdot 3} = \frac{-8 \pm 22}{6}.$

This gives us two roots:

$x = \frac{14}{6} = \frac{7}{3}$
$x = \frac{-30}{6} = -5$

Thus, the two roots are $x = \frac{7}{3}$ and $x = -5$ .

Step 2

Solve the equation $3^{2m+1} = 35 - 8(3^{m})$

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Answer

We start with the equation:

$3^{2m+1} = 35 - 8(3^m).$

Rewriting $3^{2m}$ in terms of $3^m$ yields:

$3^{2m} = (3^m)^2.$

Substituting $x = 3^m$ , the equation becomes:

$3x^2 + 8x - 35 = 0.$

From the previous steps, we know this factors to:

$(3x - 7)(x + 5) = 0.$

The solutions for $x$ are $x = \frac{7}{3}$ and $x = -5$ . Since $x = 3^m$ and must be positive, we only consider $x = \frac{7}{3}$ .

Therefore, we can express $3^m$ as:

$3^m = \frac{7}{3}.$

Taking logarithms gives:

$m = \log_3 \left(\frac{7}{3}\right) = \log_3 7 - \log_3 3 = \log_3 7 - 1.$

This is in the form $m = \log_b p - q$ , where $b = 3$ , $p = 7$ , and $q = 1$ .

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Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$ - Leaving Cert Mathematics - Question b - 2021

Worked Solution & Example Answer:Given that $f(x) = 3x^2 + 8x - 35$, where $x \in \mathbb{R}$, find the two roots of $f(x) = 0$ - Leaving Cert Mathematics - Question b - 2021

Find the two roots of $f(x) = 0$

Solve the equation $3^{2m+1} = 35 - 8(3^{m})$

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