Example Question - standard form
Here are examples of questions we've helped users solve.
Converting a Number to Scientific Notation and Floating-Point Notation
<p>For scientific notation, move the decimal point in 21.79 until there is one non-zero digit to the left of the decimal point:</p> <p>\(2.179 \times 10^1\)</p> <p>For floating-point notation, this is the same as the original number:</p> <p>\(21.79\)</p>
Solving a Quadratic Equation with the Quadratic Formula
To solve the equation \( x - (7/x) + 4 = 0 \), we can multiply through by x to clear the fraction: \[ x^2 - 7 + 4x = 0 \] Rearrange the terms to get a quadratic equation in standard form: \[ x^2 + 4x - 7 = 0 \] This quadratic equation doesn't factor nicely, so we'll need to use the quadratic formula to find the solutions for x. The quadratic formula is given by: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] For our equation, \( a = 1 \), \( b = 4 \), and \( c = -7 \), so we substitute these values into the formula: \[ x = \frac{{-4 \pm \sqrt{{4^2 - 4(1)(-7)}}}}{2(1)} \] \[ x = \frac{{-4 \pm \sqrt{{16 + 28}}}}{2} \] \[ x = \frac{{-4 \pm \sqrt{{44}}}}{2} \] \[ x = \frac{{-4 \pm 2\sqrt{{11}}}}{2} \] Now simplify by dividing all terms by 2: \[ x = -2 \pm \sqrt{{11}} \] Since the instructions say to list each solution only once and use a comma to separate the answers if needed, the complete and simplified answer is: \[ x = -2 + \sqrt{{11}}, -2 - \sqrt{{11}} \]
Circle Equation with Center and Radius
The equation given in the image is for a circle, and it is written in the standard form: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center of the circle, and \(r\) is the radius. The given equation is: \[ (x - 3)^2 + (y - 1)^2 = 4 \] From this equation, we can directly read the center and the radius of the circle: Center \((h, k)\) is \((3, 1)\), as it's the point you get by undoing the sign change in the brackets. Radius \(r\) is \(\sqrt{4}\), which is \(2\), as the radius squared \(r^2\) equals \(4\). So the center of the circle is at the point \((3, 1)\), and the radius is \(2\). To represent this graphically, you would plot the center at point \((3, 1)\) on a Cartesian plane and draw a circle around this point with a radius of \(2\) units, ensuring that all points on the circumference of the circle are \(2\) units away from the center.
Solving a Complex Fraction to Standard Form
The provided expression is a complex fraction: \[ \frac{8 - i}{3 - 2i} \] To write this in the standard form \( a + bi \), where \( a \) and \( b \) are real numbers, you must remove the imaginary part from the denominator. You can do this by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(3 - 2i\) is \(3 + 2i\). Multiplying the numerator and the denominator by \(3 + 2i\) gives us: \[ \frac{8 - i}{3 - 2i} \times \frac{3 + 2i}{3 + 2i} \] This multiplication will give us: \[ \frac{(8 - i)(3 + 2i)}{(3 - 2i)(3 + 2i)} \] Expand the numerator and the denominator: \[ \frac{24 + 16i - 3i - 2i^2}{9 - 6i + 6i - 4i^2} \] Since \(i^2 = -1\), this simplifies to: \[ \frac{24 + 13i - 2(-1)}{9 - 4(-1)} \] \[ \frac{24 + 13i + 2}{9 + 4} \] \[ \frac{26 + 13i}{13} \] \[ 2 + i \] So in the form \( a + bi \), \( a = 2 \) and \( b = 1 \). Therefore, the value of \( a \) is 2, which corresponds to option A.
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