Find the quadratic equation whose roots are x = −3 and x = 1/2.

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Find the quadratic equation whose roots are $x = -3$ and $x = \frac{1}{2}$ .

Calculate the sum and product of the roots.
Let the roots be $r_1 = -3$ and $r_2 = \frac{1}{2}$ .
Sum of roots: $S = r_1 + r_2 = -3 + \frac{1}{2} = -\frac{6}{2} + \frac{1}{2} = -\frac{5}{2}$ .
Product of roots: $P = r_1 \times r_2 = (-3) \times \frac{1}{2} = -\frac{3}{2}$ .
Form the quadratic equation using the formula $x^2 - Sx + P = 0$ .
$x^2 - \left(-\frac{5}{2}\right)x + \left(-\frac{3}{2}\right) = 0$ $x^2 + \frac{5}{2}x - \frac{3}{2} = 0$
Multiply the entire equation by 2 to clear the denominators.
$2(x^2) + 2\left(\frac{5}{2}x\right) - 2\left(\frac{3}{2}\right) = 2(0)$ $2x^2 + 5x - 3 = 0$
The correct option is B.
$\boxed{\text{B. } 2x^2 + 5x - 3 = 0}$
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Calculate $b$ in the figure below if $\hat{B}=60^\circ$ , $c=9$ cm and $a=11$ cm.
Identify the given values and the appropriate formula.
We are given two sides ( $a=11$ cm, $c=9$ cm) and the included angle ( $\hat{B}=60^\circ$ ). We need to find the third side $b$ . The Cosine Rule is suitable here:
$b^2 = a^2 + c^2 - 2ac \cos B$
Substitute the values into the Cosine Rule.
$b^2 = (11)^2 + (9)^2 - 2(11)(9) \cos 60^\circ$
Recall that $\cos 60^\circ = \frac{1}{2}$ .
$b^2 = 121 + 81 - 2(11)(9)\left(\frac{1}{2}\right)$ $b^2 = 202 - (11)(9)$ $b^2 = 202 - 99$ $b^2 = 103$
Solve for $b$ .
$b = \sqrt{103}$ $b \approx 10.1488 \text{ cm}$
Rounding to two decimal places, $b \approx 10.15$ cm.
The correct option is B.
$\boxed{\text{B. } 10.15\text{cm}}$
:
Calculate the length of a chord of a circle of radius 5cm if the perpendicular distance from the center to the chord is 3cm.
Visualize the geometry and identify the relevant right-angled triangle.
Let the radius be $r=5$ cm. Let the perpendicular distance from the center to the chord be $d=3$ cm. Let half the length of the chord be $x$ . The radius, the perpendicular distance, and half the chord form a right-angled triangle, with the radius as the hypotenuse.
Apply the Pythagorean theorem.
$x^2 + d^2 = r^2$ $x^2 + (3)^2 = (5)^2$ $x^2 + 9 = 25$ $x^2 = 25 - 9$ $x^2 = 16$
Solve for $x$ and then find the full length of the chord.
$x = \sqrt{16}$ $x = 4 \text{ cm}$
The length of the full chord is $2x$ .
$\text{Chord length} = 2 \times 4 = 8 \text{ cm}$
The correct option is A.
$\boxed{\text{A. } 8\text{cm}}$
:
A sector of a circle of 8cm has $90^\circ$ at the centre. Calculate its area.
Identify the given values.
Radius $r = 8$ cm.
Central angle $\theta = 90^\circ$ .
Use the formula for the area of a sector.
The area of a sector is given by $A = \frac{\theta}{360^\circ} \times \pi r^2$ .
$A = \frac{90^\circ}{360^\circ} \times \pi (8\text{cm})^2$ $A = \frac{1}{4} \times \pi (64\text{cm}^2)$ $A = 16\pi \text{ cm}^2$
Calculate the numerical value and round.
Using $\pi \approx 3.14159$ :
$A = 16 \times 3.14159 \text{ cm}^2$ $A \approx 50.26544 \text{ cm}^2$
Rounding to two decimal places, $A \approx 50.27$ cm $^2$ .
The correct option is D.
$\boxed{\text{D. } 50.27\text{cm}^2}$
:
In the figure above, AD//GE, /BF/ = /CF/ and $\angle FBC = 55^\circ$ . Find $\angle BFC$ .
Identify the type of triangle formed by B, F, and C.
Given that /BF/ = /CF/, triangle BFC is an isosceles triangle.
Use the property of isosceles triangles.
In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, $\angle FBC = \angle FCB$ .
Given $\angle FBC = 55^\circ$ , so $\angle FCB = 55^\circ$ .
Use the sum of angles in a triangle.
The sum of angles in $\triangle BFC$ is $180^\circ$ .
$\angle BFC + \angle FBC + \angle FCB = 180^\circ$ $\angle BFC + 55^\circ + 55^\circ = 180^\circ$ $\angle BFC + 110^\circ = 180^\circ$ $\angle BFC = 180^\circ - 110^\circ$ $\angle BFC = 70^\circ$
The information AD//GE is not needed to solve for $\angle BFC$ .
The correct option is B.
$\boxed{\text{B. } 70^\circ}$
:
The radii of two spheres are in the ratio 2:3. If the area of the smaller sphere is 32cm $^2$ , calculate the area of the bigger sphere.
Set up the ratio of the radii.
Let $r_1$ be the radius of the smaller sphere and $r_2$ be the radius of the bigger sphere.
$\frac{r_1}{r_2} = \frac{2}{3}$
Relate the ratio of radii to the ratio of surface areas.
The surface area of a sphere is given by $A = 4\pi r^2$ .
Let $A_1$ be the area of the smaller sphere and $A_2$ be the area of the bigger sphere.
The ratio of their areas is:
$\frac{A_1}{A_2} = \frac{4\pi r_1^2}{4\pi r_2^2} = \frac{r_1^2}{r_2^2} = \left(\frac{r_1}{r_2}\right)^2$
Substitute the given ratio and area to find the unknown area.
$\frac{A_1}{A_2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$
Given $A_1 = 32$ cm $^2$ .
$\frac{32}{A_2} = \frac{4}{9}$
Cross-multiply to solve for $A_2$ :
$4 A_2 = 32 \times 9$ $A_2 = \frac{32 \times 9}{4}$ $A_2 = 8 \times 9$ $A_2 = 72 \text{ cm}^2$
The correct option is A.
$\boxed{\text{A. } 72\text{cm}^2}$

Find the quadratic equation whose roots are x = -3 and x = (1)/(2).

Answer

Step-by-step solution

Calculate the sum and product of the roots.

Form the quadratic equation using the formula $x^2 - Sx + P = 0$ .

Multiply the entire equation by 2 to clear the denominators.

Identify the given values and the appropriate formula.

Substitute the values into the Cosine Rule.

Solve for $b$ .

Visualize the geometry and identify the relevant right-angled triangle.

Apply the Pythagorean theorem.

Solve for $x$ and then find the full length of the chord.

Identify the given values.

Use the formula for the area of a sector.

Calculate the numerical value and round.

Identify the type of triangle formed by B, F, and C.

Use the property of isosceles triangles.

Use the sum of angles in a triangle.

Set up the ratio of the radii.

Relate the ratio of radii to the ratio of surface areas.

Substitute the given ratio and area to find the unknown area.

Try one yourself

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More Mathematics Questions

Find the quadratic equation whose roots are x = -3 and x = (1)/(2).

Answer

Step-by-step solution

Calculate the sum and product of the roots.

Form the quadratic equation using the formula x2−Sx+P=0x^2 - Sx + P = 0x2−Sx+P=0.

Multiply the entire equation by 2 to clear the denominators.

Identify the given values and the appropriate formula.

Substitute the values into the Cosine Rule.

Solve for bbb.

Visualize the geometry and identify the relevant right-angled triangle.

Apply the Pythagorean theorem.

Solve for xxx and then find the full length of the chord.

Identify the given values.

Use the formula for the area of a sector.

Calculate the numerical value and round.

Identify the type of triangle formed by B, F, and C.

Use the property of isosceles triangles.

Use the sum of angles in a triangle.

Set up the ratio of the radii.

Relate the ratio of radii to the ratio of surface areas.

Substitute the given ratio and area to find the unknown area.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Form the quadratic equation using the formula $x^2 - Sx + P = 0$ .

Solve for $b$ .

Solve for $x$ and then find the full length of the chord.