Given the quadratic equation 2x2 - 4x + 7 = 0. Calculate the value of the discriminant and determine the…

Question

Given the quadratic equation 2x2 - 4x + 7 = 0. Calculate the value of the discriminant and determine the nature of roots. Determine the numerical value(s) of k for which the equation x2 + 6x = k will have real and equal roots.

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Here are the solutions for the questions:

5.1.1

To calculate the value of the discriminant for the quadratic equation $2x^2 - 4x + 7 = 0$ .

Identify the coefficients $a, b, c$.
For $2x^2 - 4x + 7 = 0$ , we have $a=2$ , $b=-4$ , $c=7$ .
Use the discriminant formula $\Delta = b^2 - 4ac$ .
$\Delta = (-4)^2 - 4(2)(7)$ $\Delta = 16 - 56$ $\Delta = -40$
The value of the discriminant is $-40$ .
$\boxed{\Delta = -40}$
5.1.2

To determine the nature of the roots based on the discriminant.
Analyze the value of the discriminant.
From 5.1.1, $\Delta = -40$ .
Conclude the nature of the roots.
Since the discriminant $\Delta < 0$ , the roots are non-real and unequal.
$\boxed{\text{Non-real and unequal roots}}$
5.2

To determine the numerical value(s) of $k$ for which the equation $x^2 + 6x = k$ will have real and equal roots.
Rewrite the equation in standard quadratic form $ax^2+bx+c=0$ .
$x^2 + 6x - k = 0$
Identify the coefficients $a, b, c$.
For $x^2 + 6x - k = 0$ , we have $a=1$ , $b=6$ , $c=-k$ .
Apply the condition for real and equal roots, which is $\Delta = b^2 - 4ac = 0$ .
$(6)^2 - 4(1)(-k) = 0$ $36 + 4k = 0$
Solve for $k$ .
$4k = -36$ $k = \frac{-36}{4}$ $k = -9$
The numerical value of $k$ is $-9$ .
$\boxed{k = -9}$
6.1.1

To simplify $625^{\frac{1}{4}}$ without using a calculator.
Express the base as a power.
We know that $625 = 5 \times 5 \times 5 \times 5 = 5^4$ .
Substitute and simplify using exponent rules.
$625^{\frac{1}{4}} = (5^4)^{\frac{1}{4}}$ $= 5^{4 \times \frac{1}{4}}$ $= 5^1$ $= 5$
The simplified value is 5.
$\boxed{5}$
6.1.2

To simplify $\frac{\sqrt{2^{2x} \cdot 4}}{\sqrt[3]{8^x}}$ without using a calculator.
Simplify the numerator $\sqrt{2^{2x} \cdot 4}$ .
$\sqrt{2^{2x} \cdot 4} = \sqrt{2^{2x} \cdot 2^2}$ $= \sqrt{2^{2x+2}}$ $= (2^{2x+2})^{\frac{1}{2}}$ $= 2^{\frac{1}{2}(2x+2)}$ $= 2^{x+1}$
Simplify the denominator $\sqrt[3]{8^x}$ .
$\sqrt[3]{8^x} = \sqrt[3]{(2^3)^x}$ $= \sqrt[3]{2^{3x}}$ $= (2^{3x})^{\frac{1}{3}}$ $= 2^{\frac{1}{3}(3x)}$ $= 2^x$
Divide the simplified numerator by the simplified denominator.
$\frac{2^{x+1}}{2^x} = 2^{(x+1)-x}$ $= 2^1$ $= 2$
The simplified expression is 2.
$\boxed{2}$

Given the quadratic equation 2x2 − 4x + 7 = 0. Calculate the value of the discriminant and determine the nature of roots. Determine the numerical value(s) of k for which the equation x2 + 6x = k will have real and equal roots.

Step-by-step solution

5.1.1

Identify the coefficients $a, b, c$.

Use the discriminant formula $\Delta = b^2 - 4ac$ .

5.1.2

Analyze the value of the discriminant.

Conclude the nature of the roots.

5.2

Rewrite the equation in standard quadratic form $ax^2+bx+c=0$ .

Identify the coefficients $a, b, c$.

Apply the condition for real and equal roots, which is $\Delta = b^2 - 4ac = 0$ .

Solve for $k$ .

6.1.1

Express the base as a power.

Substitute and simplify using exponent rules.

6.1.2

Simplify the numerator $\sqrt{2^{2x} \cdot 4}$ .

Simplify the denominator $\sqrt[3]{8^x}$ .

Divide the simplified numerator by the simplified denominator.

Still stuck on this one?

More Mathematics Questions

Given the quadratic equation 2x2 − 4x + 7 = 0. Calculate the value of the discriminant and determine the nature of roots. Determine the numerical value(s) of k for which the equation x2 + 6x = k will have real and equal roots.

Step-by-step solution

5.1.1

Identify the coefficients $a, b, c$.

Use the discriminant formula Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac.

5.1.2

Analyze the value of the discriminant.

Conclude the nature of the roots.

5.2

Rewrite the equation in standard quadratic form ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0.

Identify the coefficients $a, b, c$.

Apply the condition for real and equal roots, which is Δ=b2−4ac=0\Delta = b^2 - 4ac = 0Δ=b2−4ac=0.

Solve for kkk.

6.1.1

Express the base as a power.

Substitute and simplify using exponent rules.

6.1.2

Simplify the numerator 22x⋅4\sqrt{2^{2x} \cdot 4}22x⋅4​.

Simplify the denominator 8x3\sqrt[3]{8^x}38x​.

Divide the simplified numerator by the simplified denominator.

Still stuck on this one?

More Mathematics Questions

Use the discriminant formula $\Delta = b^2 - 4ac$ .

Rewrite the equation in standard quadratic form $ax^2+bx+c=0$ .

Apply the condition for real and equal roots, which is $\Delta = b^2 - 4ac = 0$ .

Solve for $k$ .

Simplify the numerator $\sqrt{2^{2x} \cdot 4}$ .

Simplify the denominator $\sqrt[3]{8^x}$ .